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<li class="chapter" data-level="1" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i><b>1</b> Introduction</a></li>
<li class="chapter" data-level="2" data-path="analysis-of-variance.html"><a href="analysis-of-variance.html"><i class="fa fa-check"></i><b>2</b> Analysis of Variance</a><ul>
<li class="chapter" data-level="2.1" data-path="analysis-of-variance.html"><a href="analysis-of-variance.html#one-way-design"><i class="fa fa-check"></i><b>2.1</b> One-Way Design</a><ul>
<li class="chapter" data-level="2.1.1" data-path="analysis-of-variance.html"><a href="analysis-of-variance.html#decomposition-of-sums-of-squares"><i class="fa fa-check"></i><b>2.1.1</b> Decomposition of Sums of Squares</a></li>
</ul></li>
<li class="chapter" data-level="2.2" data-path="analysis-of-variance.html"><a href="analysis-of-variance.html#computational-formulas"><i class="fa fa-check"></i><b>2.2</b> Computational Formulas</a></li>
<li class="chapter" data-level="2.3" data-path="analysis-of-variance.html"><a href="analysis-of-variance.html#randomized-complete-block-design"><i class="fa fa-check"></i><b>2.3</b> Randomized Complete Block Design</a><ul>
<li class="chapter" data-level="2.3.1" data-path="analysis-of-variance.html"><a href="analysis-of-variance.html#decomposition-of-sums-of-squares-1"><i class="fa fa-check"></i><b>2.3.1</b> Decomposition of Sums of Squares</a></li>
<li class="chapter" data-level="2.3.2" data-path="analysis-of-variance.html"><a href="analysis-of-variance.html#computational-formulae"><i class="fa fa-check"></i><b>2.3.2</b> Computational Formulae</a></li>
<li class="chapter" data-level="2.3.3" data-path="analysis-of-variance.html"><a href="analysis-of-variance.html#anova-rcbd-cross-products"><i class="fa fa-check"></i><b>2.3.3</b> RCBD Cross Products</a></li>
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<li class="chapter" data-level="3" data-path="bernoulli-distribution.html"><a href="bernoulli-distribution.html"><i class="fa fa-check"></i><b>3</b> Bernoulli Distribution</a><ul>
<li class="chapter" data-level="3.1" data-path="bernoulli-distribution.html"><a href="bernoulli-distribution.html#probability-mass-function"><i class="fa fa-check"></i><b>3.1</b> Probability Mass Function</a></li>
<li class="chapter" data-level="3.2" data-path="bernoulli-distribution.html"><a href="bernoulli-distribution.html#cumulative-mass-function"><i class="fa fa-check"></i><b>3.2</b> Cumulative Mass Function</a></li>
<li class="chapter" data-level="3.3" data-path="bernoulli-distribution.html"><a href="bernoulli-distribution.html#expected-values"><i class="fa fa-check"></i><b>3.3</b> Expected Values</a></li>
<li class="chapter" data-level="3.4" data-path="bernoulli-distribution.html"><a href="bernoulli-distribution.html#moment-generating-function"><i class="fa fa-check"></i><b>3.4</b> Moment Generating Function</a></li>
<li class="chapter" data-level="3.5" data-path="bernoulli-distribution.html"><a href="bernoulli-distribution.html#theorems-for-the-bernoulli-distribution"><i class="fa fa-check"></i><b>3.5</b> Theorems for the Bernoulli Distribution</a><ul>
<li class="chapter" data-level="3.5.1" data-path="bernoulli-distribution.html"><a href="bernoulli-distribution.html#validity-of-the-distribution"><i class="fa fa-check"></i><b>3.5.1</b> Validity of the Distribution</a></li>
<li class="chapter" data-level="3.5.2" data-path="bernoulli-distribution.html"><a href="bernoulli-distribution.html#sum-of-bernoulli-random-variables"><i class="fa fa-check"></i><b>3.5.2</b> Sum of Bernoulli Random Variables</a></li>
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<li class="chapter" data-level="4" data-path="binomial-distribution.html"><a href="binomial-distribution.html"><i class="fa fa-check"></i><b>4</b> Binomial Distribution</a><ul>
<li class="chapter" data-level="4.1" data-path="binomial-distribution.html"><a href="binomial-distribution.html#probability-mass-function-1"><i class="fa fa-check"></i><b>4.1</b> Probability Mass Function</a></li>
<li class="chapter" data-level="4.2" data-path="binomial-distribution.html"><a href="binomial-distribution.html#cumulative-mass-function-1"><i class="fa fa-check"></i><b>4.2</b> Cumulative Mass Function</a></li>
<li class="chapter" data-level="4.3" data-path="binomial-distribution.html"><a href="binomial-distribution.html#expected-values-1"><i class="fa fa-check"></i><b>4.3</b> Expected Values</a></li>
<li class="chapter" data-level="4.4" data-path="binomial-distribution.html"><a href="binomial-distribution.html#moment-generating-function-1"><i class="fa fa-check"></i><b>4.4</b> Moment Generating Function</a></li>
<li class="chapter" data-level="4.5" data-path="binomial-distribution.html"><a href="binomial-distribution.html#maximum-likelihood-estimator"><i class="fa fa-check"></i><b>4.5</b> Maximum Likelihood Estimator</a><ul>
<li class="chapter" data-level="4.5.1" data-path="binomial-distribution.html"><a href="binomial-distribution.html#likelihood-function"><i class="fa fa-check"></i><b>4.5.1</b> Likelihood Function</a></li>
<li class="chapter" data-level="4.5.2" data-path="binomial-distribution.html"><a href="binomial-distribution.html#log-likelihood-function"><i class="fa fa-check"></i><b>4.5.2</b> Log-likelihood Function</a></li>
<li class="chapter" data-level="4.5.3" data-path="binomial-distribution.html"><a href="binomial-distribution.html#mle-for-pi"><i class="fa fa-check"></i><b>4.5.3</b> MLE for <span class="math inline">\(\pi\)</span></a></li>
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<li class="chapter" data-level="4.6" data-path="binomial-distribution.html"><a href="binomial-distribution.html#theorems-for-the-binomial-distribution"><i class="fa fa-check"></i><b>4.6</b> Theorems for the Binomial Distribution</a><ul>
<li class="chapter" data-level="4.6.1" data-path="binomial-distribution.html"><a href="binomial-distribution.html#validity-of-the-distribution-1"><i class="fa fa-check"></i><b>4.6.1</b> Validity of the Distribution</a></li>
<li class="chapter" data-level="4.6.2" data-path="binomial-distribution.html"><a href="binomial-distribution.html#sum-of-binomial-random-variables"><i class="fa fa-check"></i><b>4.6.2</b> Sum of Binomial Random Variables</a></li>
<li class="chapter" data-level="4.6.3" data-path="binomial-distribution.html"><a href="binomial-distribution.html#sum-of-bernoulli-random-variables-1"><i class="fa fa-check"></i><b>4.6.3</b> Sum of Bernoulli Random Variables</a></li>
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<li class="chapter" data-level="5" data-path="binomial-test.html"><a href="binomial-test.html"><i class="fa fa-check"></i><b>5</b> Binomial Test</a><ul>
<li class="chapter" data-level="5.1" data-path="binomial-test.html"><a href="binomial-test.html#binomial-test-1"><i class="fa fa-check"></i><b>5.1</b> Binomial Test</a><ul>
<li class="chapter" data-level="5.1.1" data-path="binomial-test.html"><a href="binomial-test.html#test-statistic"><i class="fa fa-check"></i><b>5.1.1</b> Test Statistic</a></li>
<li class="chapter" data-level="5.1.2" data-path="binomial-test.html"><a href="binomial-test.html#definitions"><i class="fa fa-check"></i><b>5.1.2</b> Definitions</a></li>
<li class="chapter" data-level="5.1.3" data-path="binomial-test.html"><a href="binomial-test.html#hypotheses"><i class="fa fa-check"></i><b>5.1.3</b> Hypotheses</a></li>
<li class="chapter" data-level="5.1.4" data-path="binomial-test.html"><a href="binomial-test.html#decision-rule"><i class="fa fa-check"></i><b>5.1.4</b> Decision Rule</a></li>
<li class="chapter" data-level="5.1.5" data-path="binomial-test.html"><a href="binomial-test.html#power"><i class="fa fa-check"></i><b>5.1.5</b> Power</a></li>
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<li class="chapter" data-level="6" data-path="binomial-theorem.html"><a href="binomial-theorem.html"><i class="fa fa-check"></i><b>6</b> Binomial Theorem</a><ul>
<li class="chapter" data-level="6.1" data-path="binomial-theorem.html"><a href="binomial-theorem.html#traditional-proof"><i class="fa fa-check"></i><b>6.1</b> Traditional Proof</a><ul>
<li class="chapter" data-level="6.1.1" data-path="binomial-theorem.html"><a href="binomial-theorem.html#lemma-pascals-rule"><i class="fa fa-check"></i><b>6.1.1</b> Lemma: Pascal’s rule</a></li>
<li class="chapter" data-level="6.1.2" data-path="binomial-theorem.html"><a href="binomial-theorem.html#binomial-theorem-traditional"><i class="fa fa-check"></i><b>6.1.2</b> The Binomial Theorem</a></li>
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<li class="chapter" data-level="6.2" data-path="binomial-theorem.html"><a href="binomial-theorem.html#general-approach"><i class="fa fa-check"></i><b>6.2</b> General Approach</a><ul>
<li class="chapter" data-level="6.2.1" data-path="binomial-theorem.html"><a href="binomial-theorem.html#a-binomial-expansion-theorem"><i class="fa fa-check"></i><b>6.2.1</b> A Binomial Expansion Theorem</a></li>
<li class="chapter" data-level="6.2.2" data-path="binomial-theorem.html"><a href="binomial-theorem.html#corollary-binomial-theorem"><i class="fa fa-check"></i><b>6.2.2</b> Corollary: Binomial Theorem</a></li>
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<li class="chapter" data-level="6.3" data-path="binomial-theorem.html"><a href="binomial-theorem.html#other-theorems"><i class="fa fa-check"></i><b>6.3</b> Other Theorems</a><ul>
<li class="chapter" data-level="6.3.1" data-path="binomial-theorem.html"><a href="binomial-theorem.html#binomialtheorem-other-theorem-1"><i class="fa fa-check"></i><b>6.3.1</b> Theorem</a></li>
<li class="chapter" data-level="6.3.2" data-path="binomial-theorem.html"><a href="binomial-theorem.html#binomialtheorem-other-theorem-2"><i class="fa fa-check"></i><b>6.3.2</b> Theorem</a></li>
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<li class="chapter" data-level="7" data-path="central-limit-theorem.html"><a href="central-limit-theorem.html"><i class="fa fa-check"></i><b>7</b> Central Limit Theorem</a><ul>
<li class="chapter" data-level="7.1" data-path="central-limit-theorem.html"><a href="central-limit-theorem.html#central-limit-theorem"><i class="fa fa-check"></i><b>7.1</b> Theorem: Central Limit Theorem</a></li>
<li class="chapter" data-level="7.2" data-path="central-limit-theorem.html"><a href="central-limit-theorem.html#references"><i class="fa fa-check"></i><b>7.2</b> References</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="chebychevs-theorem.html"><a href="chebychevs-theorem.html"><i class="fa fa-check"></i><b>8</b> Chebychev’s Theorem</a><ul>
<li class="chapter" data-level="8.1" data-path="chebychevs-theorem.html"><a href="chebychevs-theorem.html#chebychevs-theorem-1"><i class="fa fa-check"></i><b>8.1</b> Chebychev’s Theorem</a></li>
<li class="chapter" data-level="8.2" data-path="chebychevs-theorem.html"><a href="chebychevs-theorem.html#alternate-proof-of-chebychevs-theorem"><i class="fa fa-check"></i><b>8.2</b> Alternate Proof of Chebychev’s Theorem</a></li>
<li class="chapter" data-level="8.3" data-path="chebychevs-theorem.html"><a href="chebychevs-theorem.html#chebychevs-theorem-for-absolute-deviation"><i class="fa fa-check"></i><b>8.3</b> Chebychev’s Theorem for Absolute Deviation</a></li>
</ul></li>
<li class="chapter" data-level="9" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html"><i class="fa fa-check"></i><b>9</b> Chi-Square Distribution</a><ul>
<li class="chapter" data-level="9.1" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#probability-distribution-function"><i class="fa fa-check"></i><b>9.1</b> Probability Distribution Function</a></li>
<li class="chapter" data-level="9.2" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#cumulative-distribution-function"><i class="fa fa-check"></i><b>9.2</b> Cumulative Distribution Function</a></li>
<li class="chapter" data-level="9.3" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#expected-values-2"><i class="fa fa-check"></i><b>9.3</b> Expected Values</a></li>
<li class="chapter" data-level="9.4" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#moment-generating-function-2"><i class="fa fa-check"></i><b>9.4</b> Moment Generating Function</a></li>
<li class="chapter" data-level="9.5" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#maximum-likelihood-function"><i class="fa fa-check"></i><b>9.5</b> Maximum Likelihood Function</a><ul>
<li class="chapter" data-level="9.5.1" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#likelihood-function-1"><i class="fa fa-check"></i><b>9.5.1</b> Likelihood Function</a></li>
<li class="chapter" data-level="9.5.2" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#log-likelihood-function-1"><i class="fa fa-check"></i><b>9.5.2</b> Log-likelihood Function</a></li>
<li class="chapter" data-level="9.5.3" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#mle-for-nu"><i class="fa fa-check"></i><b>9.5.3</b> MLE for <span class="math inline">\(\nu\)</span></a></li>
</ul></li>
<li class="chapter" data-level="9.6" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#theorems-for-the-chi-square-distribution"><i class="fa fa-check"></i><b>9.6</b> Theorems for the Chi-Square Distribution</a><ul>
<li class="chapter" data-level="9.6.1" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#validity-of-the-distribution-2"><i class="fa fa-check"></i><b>9.6.1</b> Validity of the Distribution</a></li>
<li class="chapter" data-level="9.6.2" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#sum-of-chi-square-random-variables"><i class="fa fa-check"></i><b>9.6.2</b> Sum of Chi-Square Random Variables</a></li>
<li class="chapter" data-level="9.6.3" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#multiple-of-a-chi-square-random-variable"><i class="fa fa-check"></i><b>9.6.3</b> Multiple of a Chi-Square Random Variable</a></li>
<li class="chapter" data-level="9.6.4" data-path="chi-square-distribution.html"><a href="chi-square-distribution.html#square-of-a-standard-normal-random-variable"><i class="fa fa-check"></i><b>9.6.4</b> Square of a Standard Normal Random Variable</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="10" data-path="combinations.html"><a href="combinations.html"><i class="fa fa-check"></i><b>10</b> Combinations</a><ul>
<li class="chapter" data-level="10.0.1" data-path="combinations.html"><a href="combinations.html#combinations-lemma"><i class="fa fa-check"></i><b>10.0.1</b> Lemma</a></li>
<li class="chapter" data-level="10.0.2" data-path="combinations.html"><a href="combinations.html#combinations-theorem"><i class="fa fa-check"></i><b>10.0.2</b> Combinations Theorem</a></li>
<li class="chapter" data-level="10.0.3" data-path="combinations.html"><a href="combinations.html#combinations-theorem-1"><i class="fa fa-check"></i><b>10.0.3</b> Theorem</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="correlation-pearsons.html"><a href="correlation-pearsons.html"><i class="fa fa-check"></i><b>11</b> Correlation (Pearson’s)</a><ul>
<li class="chapter" data-level="11.1" data-path="correlation-pearsons.html"><a href="correlation-pearsons.html#theorems-on-pearsons-correlation"><i class="fa fa-check"></i><b>11.1</b> Theorems on Pearson’s Correlation</a></li>
<li class="chapter" data-level="11.2" data-path="correlation-pearsons.html"><a href="correlation-pearsons.html#computational-formula-for-rho"><i class="fa fa-check"></i><b>11.2</b> Computational Formula for <span class="math inline">\(\rho\)</span></a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="covariance.html"><a href="covariance.html"><i class="fa fa-check"></i><b>12</b> Covariance</a><ul>
<li class="chapter" data-level="12.1" data-path="covariance.html"><a href="covariance.html#definition-of-covariance"><i class="fa fa-check"></i><b>12.1</b> Definition of Covariance</a></li>
<li class="chapter" data-level="12.2" data-path="covariance.html"><a href="covariance.html#theorems-on-covariance"><i class="fa fa-check"></i><b>12.2</b> Theorems on Covariance</a><ul>
<li class="chapter" data-level="12.2.1" data-path="covariance.html"><a href="covariance.html#theorem"><i class="fa fa-check"></i><b>12.2.1</b> Theorem</a></li>
<li class="chapter" data-level="12.2.2" data-path="covariance.html"><a href="covariance.html#covariance-theorem2"><i class="fa fa-check"></i><b>12.2.2</b> Theorem</a></li>
<li class="chapter" data-level="12.2.3" data-path="covariance.html"><a href="covariance.html#covariance-1"><i class="fa fa-check"></i><b>12.2.3</b> Covariance</a></li>
<li class="chapter" data-level="12.2.4" data-path="covariance.html"><a href="covariance.html#theorem-1"><i class="fa fa-check"></i><b>12.2.4</b> Theorem</a></li>
<li class="chapter" data-level="12.2.5" data-path="covariance.html"><a href="covariance.html#theorem-2"><i class="fa fa-check"></i><b>12.2.5</b> Theorem</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="13" data-path="experimental-designs.html"><a href="experimental-designs.html"><i class="fa fa-check"></i><b>13</b> Experimental Designs</a><ul>
<li class="chapter" data-level="13.1" data-path="experimental-designs.html"><a href="experimental-designs.html#designs-in-categorical-data-analysis"><i class="fa fa-check"></i><b>13.1</b> Designs in Categorical Data Analysis</a><ul>
<li class="chapter" data-level="13.1.1" data-path="experimental-designs.html"><a href="experimental-designs.html#case-control-study"><i class="fa fa-check"></i><b>13.1.1</b> Case Control Study</a></li>
<li class="chapter" data-level="13.1.2" data-path="experimental-designs.html"><a href="experimental-designs.html#cross-sectional-study"><i class="fa fa-check"></i><b>13.1.2</b> Cross Sectional Study</a></li>
<li class="chapter" data-level="13.1.3" data-path="experimental-designs.html"><a href="experimental-designs.html#cohort-study"><i class="fa fa-check"></i><b>13.1.3</b> Cohort Study</a></li>
<li class="chapter" data-level="13.1.4" data-path="experimental-designs.html"><a href="experimental-designs.html#randomized-study"><i class="fa fa-check"></i><b>13.1.4</b> Randomized Study</a></li>
<li class="chapter" data-level="13.1.5" data-path="experimental-designs.html"><a href="experimental-designs.html#summary-of-designs"><i class="fa fa-check"></i><b>13.1.5</b> Summary of Designs</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="14" data-path="exponential-distribution.html"><a href="exponential-distribution.html"><i class="fa fa-check"></i><b>14</b> Exponential Distribution</a><ul>
<li class="chapter" data-level="14.1" data-path="exponential-distribution.html"><a href="exponential-distribution.html#probability-distribution-function-1"><i class="fa fa-check"></i><b>14.1</b> Probability Distribution Function</a></li>
<li class="chapter" data-level="14.2" data-path="exponential-distribution.html"><a href="exponential-distribution.html#cumulative-distribution-function-1"><i class="fa fa-check"></i><b>14.2</b> Cumulative Distribution Function</a></li>
<li class="chapter" data-level="14.3" data-path="exponential-distribution.html"><a href="exponential-distribution.html#expected-values-3"><i class="fa fa-check"></i><b>14.3</b> Expected Values</a></li>
<li class="chapter" data-level="14.4" data-path="exponential-distribution.html"><a href="exponential-distribution.html#moment-generating-function-3"><i class="fa fa-check"></i><b>14.4</b> Moment Generating Function</a></li>
<li class="chapter" data-level="14.5" data-path="exponential-distribution.html"><a href="exponential-distribution.html#maximum-likelihood-estimator-1"><i class="fa fa-check"></i><b>14.5</b> Maximum Likelihood Estimator</a><ul>
<li class="chapter" data-level="14.5.1" data-path="exponential-distribution.html"><a href="exponential-distribution.html#likelihood-function-2"><i class="fa fa-check"></i><b>14.5.1</b> Likelihood Function</a></li>
<li class="chapter" data-level="14.5.2" data-path="exponential-distribution.html"><a href="exponential-distribution.html#log-likelihood-function-2"><i class="fa fa-check"></i><b>14.5.2</b> Log-likelihood Function</a></li>
<li class="chapter" data-level="14.5.3" data-path="exponential-distribution.html"><a href="exponential-distribution.html#mle-for-beta"><i class="fa fa-check"></i><b>14.5.3</b> MLE for <span class="math inline">\(\beta\)</span></a></li>
</ul></li>
<li class="chapter" data-level="14.6" data-path="exponential-distribution.html"><a href="exponential-distribution.html#theorems-for-the-exponential-distribution"><i class="fa fa-check"></i><b>14.6</b> Theorems for the Exponential Distribution</a><ul>
<li class="chapter" data-level="14.6.1" data-path="exponential-distribution.html"><a href="exponential-distribution.html#validity-of-the-distribution-3"><i class="fa fa-check"></i><b>14.6.1</b> Validity of the Distribution</a></li>
<li class="chapter" data-level="14.6.2" data-path="exponential-distribution.html"><a href="exponential-distribution.html#sum-of-exponential-random-variables"><i class="fa fa-check"></i><b>14.6.2</b> Sum of Exponential Random Variables</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="15" data-path="functions.html"><a href="functions.html"><i class="fa fa-check"></i><b>15</b> Functions</a><ul>
<li class="chapter" data-level="15.1" data-path="functions.html"><a href="functions.html#fundamental-concepts-and-definitions"><i class="fa fa-check"></i><b>15.1</b> Fundamental Concepts and Definitions</a><ul>
<li class="chapter" data-level="15.1.1" data-path="functions.html"><a href="functions.html#function-operations"><i class="fa fa-check"></i><b>15.1.1</b> Function Operations</a></li>
</ul></li>
<li class="chapter" data-level="15.2" data-path="functions.html"><a href="functions.html#identities-and-inverses"><i class="fa fa-check"></i><b>15.2</b> Identities and Inverses</a></li>
<li class="chapter" data-level="15.3" data-path="functions.html"><a href="functions.html#odd-and-even-functions"><i class="fa fa-check"></i><b>15.3</b> Odd and Even Functions</a></li>
<li class="chapter" data-level="15.4" data-path="functions.html"><a href="functions.html#theorems"><i class="fa fa-check"></i><b>15.4</b> Theorems</a><ul>
<li class="chapter" data-level="15.4.1" data-path="functions.html"><a href="functions.html#operations-on-even-functions"><i class="fa fa-check"></i><b>15.4.1</b> Operations on Even Functions</a></li>
<li class="chapter" data-level="15.4.2" data-path="functions.html"><a href="functions.html#operations-on-odd-functions"><i class="fa fa-check"></i><b>15.4.2</b> Operations on Odd Functions</a></li>
<li class="chapter" data-level="15.4.3" data-path="functions.html"><a href="functions.html#operations-on-an-odd-and-even-function"><i class="fa fa-check"></i><b>15.4.3</b> Operations on an Odd and Even Function</a></li>
<li class="chapter" data-level="15.4.4" data-path="functions.html"><a href="functions.html#derivatives-and-anti-derivatives-of-odd-functions"><i class="fa fa-check"></i><b>15.4.4</b> Derivatives and Anti-derivatives of Odd Functions</a></li>
<li class="chapter" data-level="15.4.5" data-path="functions.html"><a href="functions.html#derivatives-and-anti-derivatives-of-even-functions"><i class="fa fa-check"></i><b>15.4.5</b> Derivatives and Anti-derivatives of Even Functions</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="16" data-path="gamma-distribution.html"><a href="gamma-distribution.html"><i class="fa fa-check"></i><b>16</b> Gamma Distribution</a><ul>
<li class="chapter" data-level="16.1" data-path="gamma-distribution.html"><a href="gamma-distribution.html#probability-distribution-function-2"><i class="fa fa-check"></i><b>16.1</b> Probability Distribution Function</a></li>
<li class="chapter" data-level="16.2" data-path="gamma-distribution.html"><a href="gamma-distribution.html#cumulative-distribution-function-2"><i class="fa fa-check"></i><b>16.2</b> Cumulative Distribution Function</a></li>
<li class="chapter" data-level="16.3" data-path="gamma-distribution.html"><a href="gamma-distribution.html#expected-values-4"><i class="fa fa-check"></i><b>16.3</b> Expected Values</a></li>
<li class="chapter" data-level="16.4" data-path="gamma-distribution.html"><a href="gamma-distribution.html#moment-generating-function-4"><i class="fa fa-check"></i><b>16.4</b> Moment Generating Function</a></li>
<li class="chapter" data-level="16.5" data-path="gamma-distribution.html"><a href="gamma-distribution.html#maximum-likelihood-estimators"><i class="fa fa-check"></i><b>16.5</b> Maximum Likelihood Estimators</a><ul>
<li class="chapter" data-level="16.5.1" data-path="gamma-distribution.html"><a href="gamma-distribution.html#likelihood-function-3"><i class="fa fa-check"></i><b>16.5.1</b> Likelihood Function</a></li>
<li class="chapter" data-level="16.5.2" data-path="gamma-distribution.html"><a href="gamma-distribution.html#log-likelihood-function-3"><i class="fa fa-check"></i><b>16.5.2</b> Log-likelihood Function</a></li>
<li class="chapter" data-level="16.5.3" data-path="gamma-distribution.html"><a href="gamma-distribution.html#mle-for-alpha"><i class="fa fa-check"></i><b>16.5.3</b> MLE for <span class="math inline">\(\alpha\)</span></a></li>
<li class="chapter" data-level="16.5.4" data-path="gamma-distribution.html"><a href="gamma-distribution.html#mle-for-beta-1"><i class="fa fa-check"></i><b>16.5.4</b> MLE for <span class="math inline">\(\beta\)</span></a></li>
<li class="chapter" data-level="16.5.5" data-path="gamma-distribution.html"><a href="gamma-distribution.html#approximation-of-hatalpha-and-hatbeta"><i class="fa fa-check"></i><b>16.5.5</b> Approximation of <span class="math inline">\(\hat\alpha\)</span> and <span class="math inline">\(\hat\beta\)</span></a></li>
</ul></li>
<li class="chapter" data-level="16.6" data-path="gamma-distribution.html"><a href="gamma-distribution.html#theorems-for-the-gamma-distribution"><i class="fa fa-check"></i><b>16.6</b> Theorems for the Gamma Distribution</a><ul>
<li class="chapter" data-level="16.6.1" data-path="gamma-distribution.html"><a href="gamma-distribution.html#validity-of-the-distribution-4"><i class="fa fa-check"></i><b>16.6.1</b> Validity of the Distribution</a></li>
<li class="chapter" data-level="16.6.2" data-path="gamma-distribution.html"><a href="gamma-distribution.html#sum-of-gamma-random-variables"><i class="fa fa-check"></i><b>16.6.2</b> Sum of Gamma Random Variables</a></li>
<li class="chapter" data-level="16.6.3" data-path="gamma-distribution.html"><a href="gamma-distribution.html#sum-of-exponential-random-variables-1"><i class="fa fa-check"></i><b>16.6.3</b> Sum of Exponential Random Variables</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="17" data-path="gamma-function.html"><a href="gamma-function.html"><i class="fa fa-check"></i><b>17</b> Gamma Function</a><ul>
<li class="chapter" data-level="17.1" data-path="gamma-function.html"><a href="gamma-function.html#gamma-function-definition"><i class="fa fa-check"></i><b>17.1</b> Definition</a></li>
<li class="chapter" data-level="17.2" data-path="gamma-function.html"><a href="gamma-function.html#theorems-for-the-gamma-function"><i class="fa fa-check"></i><b>17.2</b> Theorems for the Gamma Function</a><ul>
<li class="chapter" data-level="17.2.1" data-path="gamma-function.html"><a href="gamma-function.html#gamma-function-reduction-lemma"><i class="fa fa-check"></i><b>17.2.1</b> Lemma</a></li>
<li class="chapter" data-level="17.2.2" data-path="gamma-function.html"><a href="gamma-function.html#gamma-function-reduction-relation"><i class="fa fa-check"></i><b>17.2.2</b> Theorem: The Reduction Relation</a></li>
<li class="chapter" data-level="17.2.3" data-path="gamma-function.html"><a href="gamma-function.html#corollary"><i class="fa fa-check"></i><b>17.2.3</b> Corollary</a></li>
<li class="chapter" data-level="17.2.4" data-path="gamma-function.html"><a href="gamma-function.html#theorem-3"><i class="fa fa-check"></i><b>17.2.4</b> Theorem:</a></li>
<li class="chapter" data-level="17.2.5" data-path="gamma-function.html"><a href="gamma-function.html#theorem-4"><i class="fa fa-check"></i><b>17.2.5</b> Theorem:</a></li>
</ul></li>
<li class="chapter" data-level="17.3" data-path="gamma-function.html"><a href="gamma-function.html#references-1"><i class="fa fa-check"></i><b>17.3</b> References</a></li>
</ul></li>
<li class="chapter" data-level="18" data-path="gaussian-integral.html"><a href="gaussian-integral.html"><i class="fa fa-check"></i><b>18</b> Gaussian Integral</a><ul>
<li class="chapter" data-level="18.1" data-path="gaussian-integral.html"><a href="gaussian-integral.html#theorems-for-the-gaussian-integral"><i class="fa fa-check"></i><b>18.1</b> Theorems for the Gaussian Integral</a><ul>
<li class="chapter" data-level="18.1.1" data-path="gaussian-integral.html"><a href="gaussian-integral.html#theorem-5"><i class="fa fa-check"></i><b>18.1.1</b> Theorem</a></li>
<li class="chapter" data-level="18.1.2" data-path="gaussian-integral.html"><a href="gaussian-integral.html#gaussian-integral-sqrt-pi"><i class="fa fa-check"></i><b>18.1.2</b> Theorem</a></li>
<li class="chapter" data-level="18.1.3" data-path="gaussian-integral.html"><a href="gaussian-integral.html#theorem-6"><i class="fa fa-check"></i><b>18.1.3</b> Theorem</a></li>
</ul></li>
<li class="chapter" data-level="18.2" data-path="gaussian-integral.html"><a href="gaussian-integral.html#references-2"><i class="fa fa-check"></i><b>18.2</b> References</a></li>
</ul></li>
<li class="chapter" data-level="19" data-path="geometric-distribution-0.html"><a href="geometric-distribution-0.html"><i class="fa fa-check"></i><b>19</b> Geometric Distribution (0)</a><ul>
<li class="chapter" data-level="19.1" data-path="geometric-distribution-0.html"><a href="geometric-distribution-0.html#probability-mass-function-2"><i class="fa fa-check"></i><b>19.1</b> Probability Mass Function</a></li>
<li class="chapter" data-level="19.2" data-path="geometric-distribution-0.html"><a href="geometric-distribution-0.html#cumulative-distribution-function-3"><i class="fa fa-check"></i><b>19.2</b> Cumulative Distribution Function</a></li>
<li class="chapter" data-level="19.3" data-path="geometric-distribution-0.html"><a href="geometric-distribution-0.html#expected-values-5"><i class="fa fa-check"></i><b>19.3</b> Expected Values</a></li>
</ul></li>
<li class="chapter" data-level="20" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html"><i class="fa fa-check"></i><b>20</b> Geometric Distribution (1)</a><ul>
<li class="chapter" data-level="20.1" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#probability-mass-function-3"><i class="fa fa-check"></i><b>20.1</b> Probability Mass Function</a></li>
<li class="chapter" data-level="20.2" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#cumulative-distribution-function-4"><i class="fa fa-check"></i><b>20.2</b> Cumulative Distribution Function</a></li>
<li class="chapter" data-level="20.3" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#expected-values-6"><i class="fa fa-check"></i><b>20.3</b> Expected Values</a></li>
<li class="chapter" data-level="20.4" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#moment-generating-function-5"><i class="fa fa-check"></i><b>20.4</b> Moment Generating Function</a></li>
<li class="chapter" data-level="20.5" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#maximum-likelihood-estimator-2"><i class="fa fa-check"></i><b>20.5</b> Maximum Likelihood Estimator</a><ul>
<li class="chapter" data-level="20.5.1" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#likelihood-function-4"><i class="fa fa-check"></i><b>20.5.1</b> Likelihood Function</a></li>
<li class="chapter" data-level="20.5.2" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#log-likelihood-function-4"><i class="fa fa-check"></i><b>20.5.2</b> Log-likelihood Function</a></li>
<li class="chapter" data-level="20.5.3" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#mle-for-p"><i class="fa fa-check"></i><b>20.5.3</b> MLE for <span class="math inline">\(p\)</span></a></li>
</ul></li>
<li class="chapter" data-level="20.6" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#theorems-for-the-geometric-distribution"><i class="fa fa-check"></i><b>20.6</b> Theorems for the Geometric Distribution</a><ul>
<li class="chapter" data-level="20.6.1" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#validity-of-the-distribution-5"><i class="fa fa-check"></i><b>20.6.1</b> Validity of the Distribution</a></li>
<li class="chapter" data-level="20.6.2" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#sum-of-geometric-random-variables"><i class="fa fa-check"></i><b>20.6.2</b> Sum of Geometric Random Variables</a></li>
<li class="chapter" data-level="20.6.3" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#geometric-distribution-memoryless-lemma"><i class="fa fa-check"></i><b>20.6.3</b> Lemma</a></li>
<li class="chapter" data-level="20.6.4" data-path="geometric-distribution-1.html"><a href="geometric-distribution-1.html#memoryless-property"><i class="fa fa-check"></i><b>20.6.4</b> Memoryless Property</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="21" data-path="geometric-series.html"><a href="geometric-series.html"><i class="fa fa-check"></i><b>21</b> Geometric Series</a><ul>
<li class="chapter" data-level="21.1" data-path="geometric-series.html"><a href="geometric-series.html#geometric-series-partial-and-infinite-summations"><i class="fa fa-check"></i><b>21.1</b> Partial Summation</a><ul>
<li class="chapter" data-level="21.1.1" data-path="geometric-series.html"><a href="geometric-series.html#geometric-series-partial-summation-natural"><i class="fa fa-check"></i><b>21.1.1</b> Summation for the Series on the Natural Numbers</a></li>
<li class="chapter" data-level="21.1.2" data-path="geometric-series.html"><a href="geometric-series.html#summation-for-the-series-on-the-non-negative-integers"><i class="fa fa-check"></i><b>21.1.2</b> Summation for the Series on the Non-negative Integers</a></li>
</ul></li>
<li class="chapter" data-level="21.2" data-path="geometric-series.html"><a href="geometric-series.html#proofs-of-convergence"><i class="fa fa-check"></i><b>21.2</b> Proofs of Convergence</a></li>
</ul></li>
<li class="chapter" data-level="22" data-path="hypergeometric-distribution.html"><a href="hypergeometric-distribution.html"><i class="fa fa-check"></i><b>22</b> Hypergeometric Distribution</a><ul>
<li class="chapter" data-level="22.1" data-path="hypergeometric-distribution.html"><a href="hypergeometric-distribution.html#probability-mass-function-4"><i class="fa fa-check"></i><b>22.1</b> Probability Mass Function</a></li>
<li class="chapter" data-level="22.2" data-path="hypergeometric-distribution.html"><a href="hypergeometric-distribution.html#cumulative-mass-function-2"><i class="fa fa-check"></i><b>22.2</b> Cumulative Mass Function</a></li>
<li class="chapter" data-level="22.3" data-path="hypergeometric-distribution.html"><a href="hypergeometric-distribution.html#expected-values-7"><i class="fa fa-check"></i><b>22.3</b> Expected Values</a></li>
<li class="chapter" data-level="22.4" data-path="hypergeometric-distribution.html"><a href="hypergeometric-distribution.html#moment-generating-function-6"><i class="fa fa-check"></i><b>22.4</b> Moment Generating Function</a></li>
<li class="chapter" data-level="22.5" data-path="hypergeometric-distribution.html"><a href="hypergeometric-distribution.html#theorems-for-the-hypergeometric-distribution"><i class="fa fa-check"></i><b>22.5</b> Theorems for the Hypergeometric Distribution</a><ul>
<li class="chapter" data-level="22.5.1" data-path="hypergeometric-distribution.html"><a href="hypergeometric-distribution.html#validity-of-the-distribution-6"><i class="fa fa-check"></i><b>22.5.1</b> Validity of the Distribution</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="23" data-path="integration-techniques-and-theorems.html"><a href="integration-techniques-and-theorems.html"><i class="fa fa-check"></i><b>23</b> Integration: Techniques and Theorems</a><ul>
<li class="chapter" data-level="23.1" data-path="integration-techniques-and-theorems.html"><a href="integration-techniques-and-theorems.html#elementary-theorems"><i class="fa fa-check"></i><b>23.1</b> Elementary Theorems</a><ul>
<li class="chapter" data-level="23.1.1" data-path="integration-techniques-and-theorems.html"><a href="integration-techniques-and-theorems.html#integration-even-function-about-zero"><i class="fa fa-check"></i><b>23.1.1</b> Integration of Even Functions about Zero</a></li>
<li class="chapter" data-level="23.1.2" data-path="integration-techniques-and-theorems.html"><a href="integration-techniques-and-theorems.html#corollary-1"><i class="fa fa-check"></i><b>23.1.2</b> Corollary</a></li>
<li class="chapter" data-level="23.1.3" data-path="integration-techniques-and-theorems.html"><a href="integration-techniques-and-theorems.html#integrals-of-horizontal-translations"><i class="fa fa-check"></i><b>23.1.3</b> Integrals of Horizontal Translations</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="24" data-path="logarithms.html"><a href="logarithms.html"><i class="fa fa-check"></i><b>24</b> Logarithms</a><ul>
<li class="chapter" data-level="24.1" data-path="logarithms.html"><a href="logarithms.html#the-natural-logarithm"><i class="fa fa-check"></i><b>24.1</b> The Natural Logarithm</a><ul>
<li class="chapter" data-level="24.1.1" data-path="logarithms.html"><a href="logarithms.html#logarithm-natural-logarithm"><i class="fa fa-check"></i><b>24.1.1</b> Definition: The Natural Logarithm</a></li>
<li class="chapter" data-level="24.1.2" data-path="logarithms.html"><a href="logarithms.html#natural-log-limit-to-e"><i class="fa fa-check"></i><b>24.1.2</b> Theorem</a></li>
</ul></li>
<li class="chapter" data-level="24.2" data-path="logarithms.html"><a href="logarithms.html#references-3"><i class="fa fa-check"></i><b>24.2</b> References</a></li>
</ul></li>
<li class="chapter" data-level="25" data-path="logistic-regression.html"><a href="logistic-regression.html"><i class="fa fa-check"></i><b>25</b> Logistic Regression</a><ul>
<li class="chapter" data-level="25.1" data-path="logistic-regression.html"><a href="logistic-regression.html#the-logit-transformation"><i class="fa fa-check"></i><b>25.1</b> The Logit Transformation</a><ul>
<li class="chapter" data-level="25.1.1" data-path="logistic-regression.html"><a href="logistic-regression.html#obtaining-the-logit-transformation"><i class="fa fa-check"></i><b>25.1.1</b> Obtaining the Logit Transformation</a></li>
</ul></li>
<li class="chapter" data-level="25.2" data-path="logistic-regression.html"><a href="logistic-regression.html#retrieving-the-modelled-probability"><i class="fa fa-check"></i><b>25.2</b> Retrieving the Modelled Probability</a></li>
</ul></li>
<li class="chapter" data-level="26" data-path="mantel-haenszel-test-of-linear-trend.html"><a href="mantel-haenszel-test-of-linear-trend.html"><i class="fa fa-check"></i><b>26</b> Mantel-Haenszel Test of Linear Trend</a></li>
<li class="chapter" data-level="27" data-path="mcnemar-test.html"><a href="mcnemar-test.html"><i class="fa fa-check"></i><b>27</b> McNemar Test</a><ul>
<li class="chapter" data-level="27.1" data-path="mcnemar-test.html"><a href="mcnemar-test.html#sample-size-calculations-for-paired-design"><i class="fa fa-check"></i><b>27.1</b> Sample Size Calculations for Paired Design</a><ul>
<li class="chapter" data-level="27.1.1" data-path="mcnemar-test.html"><a href="mcnemar-test.html#miettenens-sample-size-calculation"><i class="fa fa-check"></i><b>27.1.1</b> Miettenen’s Sample Size Calculation</a></li>
<li class="chapter" data-level="27.1.2" data-path="mcnemar-test.html"><a href="mcnemar-test.html#connors-sample-size-calculation"><i class="fa fa-check"></i><b>27.1.2</b> Connor’s Sample Size Calculation</a></li>
</ul></li>
<li class="chapter" data-level="27.2" data-path="mcnemar-test.html"><a href="mcnemar-test.html#power-calculation-for-paired-design"><i class="fa fa-check"></i><b>27.2</b> Power Calculation for Paired Design</a><ul>
<li class="chapter" data-level="27.2.1" data-path="mcnemar-test.html"><a href="mcnemar-test.html#power-calculation-for-miettenen-method"><i class="fa fa-check"></i><b>27.2.1</b> Power Calculation for Miettenen Method</a></li>
<li class="chapter" data-level="27.2.2" data-path="mcnemar-test.html"><a href="mcnemar-test.html#power-calculation-for-connor-method"><i class="fa fa-check"></i><b>27.2.2</b> Power Calculation for Connor Method</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="28" data-path="moments-and-moment-generating-functions.html"><a href="moments-and-moment-generating-functions.html"><i class="fa fa-check"></i><b>28</b> Moments and Moment Generating Functions</a><ul>
<li class="chapter" data-level="28.1" data-path="moments-and-moment-generating-functions.html"><a href="moments-and-moment-generating-functions.html#definitions-of-moments"><i class="fa fa-check"></i><b>28.1</b> Definitions of Moments</a><ul>
<li class="chapter" data-level="28.1.1" data-path="moments-and-moment-generating-functions.html"><a href="moments-and-moment-generating-functions.html#definition-general-definition-of-moments"><i class="fa fa-check"></i><b>28.1.1</b> Definition: General Definition of Moments</a></li>
<li class="chapter" data-level="28.1.2" data-path="moments-and-moment-generating-functions.html"><a href="moments-and-moment-generating-functions.html#definition-ordinary-moments"><i class="fa fa-check"></i><b>28.1.2</b> Definition: Ordinary Moments</a></li>
<li class="chapter" data-level="28.1.3" data-path="moments-and-moment-generating-functions.html"><a href="moments-and-moment-generating-functions.html#definition-central-moments"><i class="fa fa-check"></i><b>28.1.3</b> Definition: Central Moments</a></li>
</ul></li>
<li class="chapter" data-level="28.2" data-path="moments-and-moment-generating-functions.html"><a href="moments-and-moment-generating-functions.html#moment-generating-functions"><i class="fa fa-check"></i><b>28.2</b> Moment Generating Functions</a><ul>
<li class="chapter" data-level="28.2.1" data-path="moments-and-moment-generating-functions.html"><a href="moments-and-moment-generating-functions.html#moment-definition-mgf"><i class="fa fa-check"></i><b>28.2.1</b> Definition: Moment Generating Function</a></li>
<li class="chapter" data-level="28.2.2" data-path="moments-and-moment-generating-functions.html"><a href="moments-and-moment-generating-functions.html#theorem-extraction-of-moments-from-moment-generating-functions"><i class="fa fa-check"></i><b>28.2.2</b> Theorem: Extraction of Moments from Moment Generating Functions</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="29" data-path="multinomial-distribution.html"><a href="multinomial-distribution.html"><i class="fa fa-check"></i><b>29</b> Multinomial Distribution</a><ul>
<li class="chapter" data-level="29.1" data-path="multinomial-distribution.html"><a href="multinomial-distribution.html#cumulative-distribution-function-5"><i class="fa fa-check"></i><b>29.1</b> Cumulative Distribution Function</a></li>
<li class="chapter" data-level="29.2" data-path="multinomial-distribution.html"><a href="multinomial-distribution.html#expected-values-8"><i class="fa fa-check"></i><b>29.2</b> Expected Values</a></li>
</ul></li>
<li class="chapter" data-level="30" data-path="normal-distribution.html"><a href="normal-distribution.html"><i class="fa fa-check"></i><b>30</b> Normal Distribution</a><ul>
<li class="chapter" data-level="30.1" data-path="normal-distribution.html"><a href="normal-distribution.html#probability-distribution-function-3"><i class="fa fa-check"></i><b>30.1</b> Probability Distribution Function</a></li>
<li class="chapter" data-level="30.2" data-path="normal-distribution.html"><a href="normal-distribution.html#cumulative-distribution-function-6"><i class="fa fa-check"></i><b>30.2</b> Cumulative Distribution Function</a></li>
<li class="chapter" data-level="30.3" data-path="normal-distribution.html"><a href="normal-distribution.html#expected-values-9"><i class="fa fa-check"></i><b>30.3</b> Expected Values</a></li>
<li class="chapter" data-level="30.4" data-path="normal-distribution.html"><a href="normal-distribution.html#moment-generating-function-7"><i class="fa fa-check"></i><b>30.4</b> Moment Generating Function</a></li>
<li class="chapter" data-level="30.5" data-path="normal-distribution.html"><a href="normal-distribution.html#maximum-likelihood-estimators-1"><i class="fa fa-check"></i><b>30.5</b> Maximum Likelihood Estimators</a><ul>
<li class="chapter" data-level="30.5.1" data-path="normal-distribution.html"><a href="normal-distribution.html#likelihood-function-5"><i class="fa fa-check"></i><b>30.5.1</b> Likelihood Function</a></li>
<li class="chapter" data-level="30.5.2" data-path="normal-distribution.html"><a href="normal-distribution.html#log-likelihood-function-5"><i class="fa fa-check"></i><b>30.5.2</b> Log-likelihood Function</a></li>
<li class="chapter" data-level="30.5.3" data-path="normal-distribution.html"><a href="normal-distribution.html#mle-for-mu"><i class="fa fa-check"></i><b>30.5.3</b> MLE for <span class="math inline">\(\mu\)</span></a></li>
<li class="chapter" data-level="30.5.4" data-path="normal-distribution.html"><a href="normal-distribution.html#mle-for-sigma2"><i class="fa fa-check"></i><b>30.5.4</b> MLE for <span class="math inline">\(\sigma^2\)</span></a></li>
</ul></li>
<li class="chapter" data-level="30.6" data-path="normal-distribution.html"><a href="normal-distribution.html#theorems-for-the-normal-distribution"><i class="fa fa-check"></i><b>30.6</b> Theorems for the Normal Distribution</a><ul>
<li class="chapter" data-level="30.6.1" data-path="normal-distribution.html"><a href="normal-distribution.html#validity-of-the-distribution-polar-coordinates"><i class="fa fa-check"></i><b>30.6.1</b> Validity of the Distribution (Polar Coordinates)</a></li>
<li class="chapter" data-level="30.6.2" data-path="normal-distribution.html"><a href="normal-distribution.html#validity-of-the-distribution-gamma-function"><i class="fa fa-check"></i><b>30.6.2</b> Validity of the Distribution (Gamma Function)</a></li>
<li class="chapter" data-level="30.6.3" data-path="normal-distribution.html"><a href="normal-distribution.html#multiple-of-a-normal-random-variable"><i class="fa fa-check"></i><b>30.6.3</b> Multiple of a Normal Random Variable</a></li>
<li class="chapter" data-level="30.6.4" data-path="normal-distribution.html"><a href="normal-distribution.html#weighted-sum-of-normal-random-variables"><i class="fa fa-check"></i><b>30.6.4</b> Weighted Sum of Normal Random Variables</a></li>
<li class="chapter" data-level="30.6.5" data-path="normal-distribution.html"><a href="normal-distribution.html#sum-of-normal-random-variables"><i class="fa fa-check"></i><b>30.6.5</b> Sum of Normal Random Variables</a></li>
<li class="chapter" data-level="30.6.6" data-path="normal-distribution.html"><a href="normal-distribution.html#standard-normal-transformation"><i class="fa fa-check"></i><b>30.6.6</b> Standard Normal Transformation</a></li>
<li class="chapter" data-level="30.6.7" data-path="normal-distribution.html"><a href="normal-distribution.html#distribution-of-bar-x"><i class="fa fa-check"></i><b>30.6.7</b> Distribution of <span class="math inline">\(\bar X\)</span></a></li>
<li class="chapter" data-level="30.6.8" data-path="normal-distribution.html"><a href="normal-distribution.html#square-of-a-standard-normal-random-variable-1"><i class="fa fa-check"></i><b>30.6.8</b> Square of a Standard Normal Random Variable</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="31" data-path="poisson-distribution.html"><a href="poisson-distribution.html"><i class="fa fa-check"></i><b>31</b> Poisson Distribution</a><ul>
<li class="chapter" data-level="31.1" data-path="poisson-distribution.html"><a href="poisson-distribution.html#probability-mass-function-5"><i class="fa fa-check"></i><b>31.1</b> Probability Mass Function</a></li>
<li class="chapter" data-level="31.2" data-path="poisson-distribution.html"><a href="poisson-distribution.html#cumulative-mass-function-3"><i class="fa fa-check"></i><b>31.2</b> Cumulative Mass Function</a></li>
<li class="chapter" data-level="31.3" data-path="poisson-distribution.html"><a href="poisson-distribution.html#expected-values-10"><i class="fa fa-check"></i><b>31.3</b> Expected Values</a></li>
<li class="chapter" data-level="31.4" data-path="poisson-distribution.html"><a href="poisson-distribution.html#moment-generating-function-8"><i class="fa fa-check"></i><b>31.4</b> Moment Generating Function</a></li>
<li class="chapter" data-level="31.5" data-path="poisson-distribution.html"><a href="poisson-distribution.html#maximum-likelihood-estimator-3"><i class="fa fa-check"></i><b>31.5</b> Maximum Likelihood Estimator</a><ul>
<li class="chapter" data-level="31.5.1" data-path="poisson-distribution.html"><a href="poisson-distribution.html#likelihood-function-6"><i class="fa fa-check"></i><b>31.5.1</b> Likelihood Function</a></li>
<li class="chapter" data-level="31.5.2" data-path="poisson-distribution.html"><a href="poisson-distribution.html#log-likelihood"><i class="fa fa-check"></i><b>31.5.2</b> Log-likelihood</a></li>
<li class="chapter" data-level="31.5.3" data-path="poisson-distribution.html"><a href="poisson-distribution.html#mle-for-lambda"><i class="fa fa-check"></i><b>31.5.3</b> MLE for <span class="math inline">\(\lambda\)</span></a></li>
</ul></li>
<li class="chapter" data-level="31.6" data-path="poisson-distribution.html"><a href="poisson-distribution.html#theorems-for-the-poisson-distribution"><i class="fa fa-check"></i><b>31.6</b> Theorems for the Poisson Distribution</a><ul>
<li class="chapter" data-level="31.6.1" data-path="poisson-distribution.html"><a href="poisson-distribution.html#derivation-of-the-poisson-distribution"><i class="fa fa-check"></i><b>31.6.1</b> Derivation of the Poisson Distribution</a></li>
<li class="chapter" data-level="31.6.2" data-path="poisson-distribution.html"><a href="poisson-distribution.html#validity-of-the-distribution-7"><i class="fa fa-check"></i><b>31.6.2</b> Validity of the Distribution</a></li>
<li class="chapter" data-level="31.6.3" data-path="poisson-distribution.html"><a href="poisson-distribution.html#sum-of-poisson-random-variables"><i class="fa fa-check"></i><b>31.6.3</b> Sum of Poisson Random Variables</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="32" data-path="probability.html"><a href="probability.html"><i class="fa fa-check"></i><b>32</b> Probability</a><ul>
<li class="chapter" data-level="32.1" data-path="probability.html"><a href="probability.html#elementary-probability-concepts"><i class="fa fa-check"></i><b>32.1</b> Elementary Probability Concepts</a><ul>
<li class="chapter" data-level="32.1.1" data-path="probability.html"><a href="probability.html#definition-of-probability"><i class="fa fa-check"></i><b>32.1.1</b> Definition of Probability</a></li>
<li class="chapter" data-level="32.1.2" data-path="probability.html"><a href="probability.html#definition-conditional-probability"><i class="fa fa-check"></i><b>32.1.2</b> Definition: Conditional Probability</a></li>
<li class="chapter" data-level="32.1.3" data-path="probability.html"><a href="probability.html#definition-independence"><i class="fa fa-check"></i><b>32.1.3</b> Definition: Independence</a></li>
<li class="chapter" data-level="32.1.4" data-path="probability.html"><a href="probability.html#theorem-multiplicative-law-of-probability"><i class="fa fa-check"></i><b>32.1.4</b> Theorem: Multiplicative Law of Probability</a></li>
<li class="chapter" data-level="32.1.5" data-path="probability.html"><a href="probability.html#corollary-2"><i class="fa fa-check"></i><b>32.1.5</b> Corollary</a></li>
<li class="chapter" data-level="32.1.6" data-path="probability.html"><a href="probability.html#additive-law-of-probability"><i class="fa fa-check"></i><b>32.1.6</b> Additive Law of Probability</a></li>
<li class="chapter" data-level="32.1.7" data-path="probability.html"><a href="probability.html#corollary-3"><i class="fa fa-check"></i><b>32.1.7</b> Corollary</a></li>
<li class="chapter" data-level="32.1.8" data-path="probability.html"><a href="probability.html#theorem-law-of-complements"><i class="fa fa-check"></i><b>32.1.8</b> Theorem: Law of Complements</a></li>
<li class="chapter" data-level="32.1.9" data-path="probability.html"><a href="probability.html#definition-partition-of-a-sample-space"><i class="fa fa-check"></i><b>32.1.9</b> Definition: Partition of a Sample Space</a></li>
<li class="chapter" data-level="32.1.10" data-path="probability.html"><a href="probability.html#definition-decomposition"><i class="fa fa-check"></i><b>32.1.10</b> Definition: Decomposition</a></li>
<li class="chapter" data-level="32.1.11" data-path="probability.html"><a href="probability.html#theorem-total-law-of-probability"><i class="fa fa-check"></i><b>32.1.11</b> Theorem: Total Law of Probability</a></li>
<li class="chapter" data-level="32.1.12" data-path="probability.html"><a href="probability.html#theorem-bayes-rule"><i class="fa fa-check"></i><b>32.1.12</b> Theorem: Bayes’ Rule</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="33" data-path="real-number-system.html"><a href="real-number-system.html"><i class="fa fa-check"></i><b>33</b> Real Number System</a><ul>
<li class="chapter" data-level="33.1" data-path="real-number-system.html"><a href="real-number-system.html#historical-note"><i class="fa fa-check"></i><b>33.1</b> Historical Note</a></li>
<li class="chapter" data-level="33.2" data-path="real-number-system.html"><a href="real-number-system.html#the-field-of-real-numbers"><i class="fa fa-check"></i><b>33.2</b> The Field of Real Numbers</a><ul>
<li class="chapter" data-level="33.2.1" data-path="real-number-system.html"><a href="real-number-system.html#definition-the-field-of-real-numbers"><i class="fa fa-check"></i><b>33.2.1</b> Definition: The Field of Real Numbers</a></li>
<li class="chapter" data-level="33.2.2" data-path="real-number-system.html"><a href="real-number-system.html#field-axioms"><i class="fa fa-check"></i><b>33.2.2</b> Field Axioms</a></li>
<li class="chapter" data-level="33.2.3" data-path="real-number-system.html"><a href="real-number-system.html#real-numbers-definition-less-than-equal-to"><i class="fa fa-check"></i><b>33.2.3</b> Definiton: Less Than (or Equal To)</a></li>
<li class="chapter" data-level="33.2.4" data-path="real-number-system.html"><a href="real-number-system.html#definition-bounded-above-and-below"><i class="fa fa-check"></i><b>33.2.4</b> Definition: Bounded Above (and Below)</a></li>
<li class="chapter" data-level="33.2.5" data-path="real-number-system.html"><a href="real-number-system.html#definition-least-upper-and-lower-bound"><i class="fa fa-check"></i><b>33.2.5</b> Definition: Least Upper (and Lower) Bound</a></li>
<li class="chapter" data-level="33.2.6" data-path="real-number-system.html"><a href="real-number-system.html#completeness-axiom"><i class="fa fa-check"></i><b>33.2.6</b> Completeness Axiom</a></li>
</ul></li>
<li class="chapter" data-level="33.3" data-path="real-number-system.html"><a href="real-number-system.html#proof-that-the-field-of-rationals-is-not-complete"><i class="fa fa-check"></i><b>33.3</b> Proof that the Field of Rationals is not Complete</a></li>
<li class="chapter" data-level="33.4" data-path="real-number-system.html"><a href="real-number-system.html#preliminary-results-in-the-field-of-real-numbers"><i class="fa fa-check"></i><b>33.4</b> Preliminary Results in the Field of Real Numbers</a><ul>
<li class="chapter" data-level="33.4.1" data-path="real-number-system.html"><a href="real-number-system.html#theorem-uniqueness-of-identities"><i class="fa fa-check"></i><b>33.4.1</b> Theorem: Uniqueness of Identities</a></li>
<li class="chapter" data-level="33.4.2" data-path="real-number-system.html"><a href="real-number-system.html#theorem-2-uniqueness-of-inverses"><i class="fa fa-check"></i><b>33.4.2</b> Theorem 2: Uniqueness of Inverses</a></li>
<li class="chapter" data-level="33.4.3" data-path="real-number-system.html"><a href="real-number-system.html#theorem-left-cancellation-law"><i class="fa fa-check"></i><b>33.4.3</b> Theorem: Left Cancellation Law</a></li>
<li class="chapter" data-level="33.4.4" data-path="real-number-system.html"><a href="real-number-system.html#corollary-right-cancellation"><i class="fa fa-check"></i><b>33.4.4</b> Corollary: Right Cancellation</a></li>
<li class="chapter" data-level="33.4.5" data-path="real-number-system.html"><a href="real-number-system.html#lemma"><i class="fa fa-check"></i><b>33.4.5</b> Lemma</a></li>
<li class="chapter" data-level="33.4.6" data-path="real-number-system.html"><a href="real-number-system.html#theorem-8"><i class="fa fa-check"></i><b>33.4.6</b> Theorem</a></li>
<li class="chapter" data-level="33.4.7" data-path="real-number-system.html"><a href="real-number-system.html#theorem-9"><i class="fa fa-check"></i><b>33.4.7</b> Theorem</a></li>
<li class="chapter" data-level="33.4.8" data-path="real-number-system.html"><a href="real-number-system.html#theorem-10"><i class="fa fa-check"></i><b>33.4.8</b> Theorem</a></li>
<li class="chapter" data-level="33.4.9" data-path="real-number-system.html"><a href="real-number-system.html#real-numbers-theorem-4-9"><i class="fa fa-check"></i><b>33.4.9</b> Theorem</a></li>
<li class="chapter" data-level="33.4.10" data-path="real-number-system.html"><a href="real-number-system.html#real-numbers-theorem-4-10"><i class="fa fa-check"></i><b>33.4.10</b> Theorem</a></li>
<li class="chapter" data-level="33.4.11" data-path="real-number-system.html"><a href="real-number-system.html#real-numbers-theorem-4-11"><i class="fa fa-check"></i><b>33.4.11</b> Corollary</a></li>
<li class="chapter" data-level="33.4.12" data-path="real-number-system.html"><a href="real-number-system.html#theorem-11"><i class="fa fa-check"></i><b>33.4.12</b> Theorem</a></li>
<li class="chapter" data-level="33.4.13" data-path="real-number-system.html"><a href="real-number-system.html#theorem-12"><i class="fa fa-check"></i><b>33.4.13</b> Theorem</a></li>
<li class="chapter" data-level="33.4.14" data-path="real-number-system.html"><a href="real-number-system.html#theorem-13"><i class="fa fa-check"></i><b>33.4.14</b> Theorem</a></li>
<li class="chapter" data-level="33.4.15" data-path="real-number-system.html"><a href="real-number-system.html#theorem-14"><i class="fa fa-check"></i><b>33.4.15</b> Theorem</a></li>
<li class="chapter" data-level="33.4.16" data-path="real-number-system.html"><a href="real-number-system.html#theorem-15"><i class="fa fa-check"></i><b>33.4.16</b> Theorem</a></li>
<li class="chapter" data-level="33.4.17" data-path="real-number-system.html"><a href="real-number-system.html#theorem-16"><i class="fa fa-check"></i><b>33.4.17</b> Theorem</a></li>
<li class="chapter" data-level="33.4.18" data-path="real-number-system.html"><a href="real-number-system.html#theorem-17"><i class="fa fa-check"></i><b>33.4.18</b> Theorem</a></li>
<li class="chapter" data-level="33.4.19" data-path="real-number-system.html"><a href="real-number-system.html#theorem-18"><i class="fa fa-check"></i><b>33.4.19</b> Theorem</a></li>
<li class="chapter" data-level="33.4.20" data-path="real-number-system.html"><a href="real-number-system.html#theorem-19"><i class="fa fa-check"></i><b>33.4.20</b> Theorem</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="34" data-path="rounding.html"><a href="rounding.html"><i class="fa fa-check"></i><b>34</b> Rounding</a><ul>
<li class="chapter" data-level="34.1" data-path="rounding.html"><a href="rounding.html#floor-next-lowest-integer"><i class="fa fa-check"></i><b>34.1</b> Floor (Next Lowest Integer)</a></li>
<li class="chapter" data-level="34.2" data-path="rounding.html"><a href="rounding.html#ceiling"><i class="fa fa-check"></i><b>34.2</b> Ceiling</a></li>
<li class="chapter" data-level="34.3" data-path="rounding.html"><a href="rounding.html#nearest-integer"><i class="fa fa-check"></i><b>34.3</b> Nearest Integer</a></li>
<li class="chapter" data-level="34.4" data-path="rounding.html"><a href="rounding.html#nearest-multiple"><i class="fa fa-check"></i><b>34.4</b> Nearest Multiple</a></li>
<li class="chapter" data-level="34.5" data-path="rounding.html"><a href="rounding.html#nearest-place-base-10"><i class="fa fa-check"></i><b>34.5</b> Nearest Place (Base 10)</a></li>
<li class="chapter" data-level="34.6" data-path="rounding.html"><a href="rounding.html#breaking-ties"><i class="fa fa-check"></i><b>34.6</b> Breaking Ties</a><ul>
<li class="chapter" data-level="34.6.1" data-path="rounding.html"><a href="rounding.html#rounding-even"><i class="fa fa-check"></i><b>34.6.1</b> Rounding Even</a></li>
<li class="chapter" data-level="34.6.2" data-path="rounding.html"><a href="rounding.html#rounding-odd"><i class="fa fa-check"></i><b>34.6.2</b> Rounding Odd</a></li>
<li class="chapter" data-level="34.6.3" data-path="rounding.html"><a href="rounding.html#round-away-from-zero"><i class="fa fa-check"></i><b>34.6.3</b> Round Away From Zero</a></li>
<li class="chapter" data-level="34.6.4" data-path="rounding.html"><a href="rounding.html#round-toward-zero"><i class="fa fa-check"></i><b>34.6.4</b> Round Toward Zero</a></li>
</ul></li>
<li class="chapter" data-level="34.7" data-path="rounding.html"><a href="rounding.html#references-4"><i class="fa fa-check"></i><b>34.7</b> References</a></li>
</ul></li>
<li class="chapter" data-level="35" data-path="sample-size-estimation.html"><a href="sample-size-estimation.html"><i class="fa fa-check"></i><b>35</b> Sample Size Estimation</a><ul>
<li class="chapter" data-level="35.1" data-path="sample-size-estimation.html"><a href="sample-size-estimation.html#solving-group-sample-sizes-using-weights"><i class="fa fa-check"></i><b>35.1</b> Solving Group Sample Sizes Using Weights</a><ul>
<li class="chapter" data-level="35.1.1" data-path="sample-size-estimation.html"><a href="sample-size-estimation.html#corollary-4"><i class="fa fa-check"></i><b>35.1.1</b> Corollary</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="36" data-path="skew-normal-distribution.html"><a href="skew-normal-distribution.html"><i class="fa fa-check"></i><b>36</b> Skew-Normal Distribution</a><ul>
<li class="chapter" data-level="36.1" data-path="skew-normal-distribution.html"><a href="skew-normal-distribution.html#preliminary-theorems"><i class="fa fa-check"></i><b>36.1</b> Preliminary Theorems</a></li>
<li class="chapter" data-level="36.2" data-path="skew-normal-distribution.html"><a href="skew-normal-distribution.html#lemma-a-symmetry-theorem"><i class="fa fa-check"></i><b>36.2</b> Lemma: A Symmetry Theorem</a></li>
<li class="chapter" data-level="36.3" data-path="skew-normal-distribution.html"><a href="skew-normal-distribution.html#lemma-1"><i class="fa fa-check"></i><b>36.3</b> Lemma</a></li>
<li class="chapter" data-level="36.4" data-path="skew-normal-distribution.html"><a href="skew-normal-distribution.html#expected-values-11"><i class="fa fa-check"></i><b>36.4</b> Expected Values</a></li>
<li class="chapter" data-level="36.5" data-path="skew-normal-distribution.html"><a href="skew-normal-distribution.html#estimation-of-lambda"><i class="fa fa-check"></i><b>36.5</b> Estimation of <span class="math inline">\(\lambda\)</span></a></li>
</ul></li>
<li class="chapter" data-level="37" data-path="somers-d.html"><a href="somers-d.html"><i class="fa fa-check"></i><b>37</b> Somers’ D</a><ul>
<li class="chapter" data-level="37.1" data-path="somers-d.html"><a href="somers-d.html#theorems-for-somers-d"><i class="fa fa-check"></i><b>37.1</b> Theorems for Somers’ <span class="math inline">\(D\)</span></a><ul>
<li class="chapter" data-level="37.1.1" data-path="somers-d.html"><a href="somers-d.html#theorem-distribution-of-somers-derived-concordance"><i class="fa fa-check"></i><b>37.1.1</b> Theorem: Distribution of Somers’ Derived Concordance</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="38" data-path="summation-chapter.html"><a href="summation-chapter.html"><i class="fa fa-check"></i><b>38</b> Summation</a><ul>
<li class="chapter" data-level="38.1" data-path="summation-chapter.html"><a href="summation-chapter.html#theorems-of-summation"><i class="fa fa-check"></i><b>38.1</b> Theorems of Summation</a><ul>
<li class="chapter" data-level="38.1.1" data-path="summation-chapter.html"><a href="summation-chapter.html#summation-theorem-1"><i class="fa fa-check"></i><b>38.1.1</b> Theorem</a></li>
<li class="chapter" data-level="38.1.2" data-path="summation-chapter.html"><a href="summation-chapter.html#summation-theorem-2"><i class="fa fa-check"></i><b>38.1.2</b> Theorem</a></li>
<li class="chapter" data-level="38.1.3" data-path="summation-chapter.html"><a href="summation-chapter.html#summation-theorem-3"><i class="fa fa-check"></i><b>38.1.3</b> Theorem</a></li>
<li class="chapter" data-level="38.1.4" data-path="summation-chapter.html"><a href="summation-chapter.html#summation-theorem-4"><i class="fa fa-check"></i><b>38.1.4</b> Theorem</a></li>
<li class="chapter" data-level="38.1.5" data-path="summation-chapter.html"><a href="summation-chapter.html#summation-theorem-5"><i class="fa fa-check"></i><b>38.1.5</b> Theorem</a></li>
<li class="chapter" data-level="38.1.6" data-path="summation-chapter.html"><a href="summation-chapter.html#summation-theorem-6"><i class="fa fa-check"></i><b>38.1.6</b> Theorem</a></li>
<li class="chapter" data-level="38.1.7" data-path="summation-chapter.html"><a href="summation-chapter.html#summation-theorem-7"><i class="fa fa-check"></i><b>38.1.7</b> Theorem</a></li>
<li class="chapter" data-level="38.1.8" data-path="summation-chapter.html"><a href="summation-chapter.html#summation-theorem-8"><i class="fa fa-check"></i><b>38.1.8</b> Theorem</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="39" data-path="analysis-of-variance-1.html"><a href="analysis-of-variance-1.html"><i class="fa fa-check"></i><b>39</b> Analysis of Variance</a><ul>
<li class="chapter" data-level="39.1" data-path="analysis-of-variance-1.html"><a href="analysis-of-variance-1.html#one-way-design-1"><i class="fa fa-check"></i><b>39.1</b> One-Way Design</a><ul>
<li class="chapter" data-level="39.1.1" data-path="analysis-of-variance-1.html"><a href="analysis-of-variance-1.html#decomposition-of-sums-of-squares-2"><i class="fa fa-check"></i><b>39.1.1</b> Decomposition of Sums of Squares</a></li>
</ul></li>
<li class="chapter" data-level="39.2" data-path="analysis-of-variance-1.html"><a href="analysis-of-variance-1.html#computational-formulas-1"><i class="fa fa-check"></i><b>39.2</b> Computational Formulas</a></li>
<li class="chapter" data-level="39.3" data-path="analysis-of-variance-1.html"><a href="analysis-of-variance-1.html#randomized-complete-block-design-1"><i class="fa fa-check"></i><b>39.3</b> Randomized Complete Block Design</a><ul>
<li class="chapter" data-level="39.3.1" data-path="analysis-of-variance-1.html"><a href="analysis-of-variance-1.html#decomposition-of-sums-of-squares-3"><i class="fa fa-check"></i><b>39.3.1</b> Decomposition of Sums of Squares</a></li>
<li class="chapter" data-level="39.3.2" data-path="analysis-of-variance-1.html"><a href="analysis-of-variance-1.html#computational-formulae-1"><i class="fa fa-check"></i><b>39.3.2</b> Computational Formulae</a></li>
<li class="chapter" data-level="39.3.3" data-path="analysis-of-variance.html"><a href="analysis-of-variance.html#anova-rcbd-cross-products"><i class="fa fa-check"></i><b>39.3.3</b> RCBD Cross Products</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="40" data-path="method-of-transformations.html"><a href="method-of-transformations.html"><i class="fa fa-check"></i><b>40</b> Method of Transformations</a><ul>
<li class="chapter" data-level="40.1" data-path="method-of-transformations.html"><a href="method-of-transformations.html#example-cauchy-distribution"><i class="fa fa-check"></i><b>40.1</b> Example: Cauchy Distribution</a></li>
</ul></li>
<li class="chapter" data-level="41" data-path="t-test.html"><a href="t-test.html"><i class="fa fa-check"></i><b>41</b> T-test</a><ul>
<li class="chapter" data-level="41.1" data-path="t-test.html"><a href="t-test.html#one-sample-t-test"><i class="fa fa-check"></i><b>41.1</b> One-Sample T-test</a><ul>
<li class="chapter" data-level="41.1.1" data-path="t-test.html"><a href="t-test.html#t-statistic"><i class="fa fa-check"></i><b>41.1.1</b> T-Statistic</a></li>
<li class="chapter" data-level="41.1.2" data-path="t-test.html"><a href="t-test.html#definitions-and-terminology"><i class="fa fa-check"></i><b>41.1.2</b> Definitions and Terminology</a></li>
<li class="chapter" data-level="41.1.3" data-path="t-test.html"><a href="t-test.html#hypotheses-1"><i class="fa fa-check"></i><b>41.1.3</b> Hypotheses</a></li>
<li class="chapter" data-level="41.1.4" data-path="t-test.html"><a href="t-test.html#decision-rule-1"><i class="fa fa-check"></i><b>41.1.4</b> Decision Rule</a></li>
<li class="chapter" data-level="41.1.5" data-path="t-test.html"><a href="t-test.html#power-1"><i class="fa fa-check"></i><b>41.1.5</b> Power</a></li>
<li class="chapter" data-level="41.1.6" data-path="t-test.html"><a href="t-test.html#confidence-interval"><i class="fa fa-check"></i><b>41.1.6</b> Confidence Interval</a></li>
</ul></li>
<li class="chapter" data-level="41.2" data-path="t-test.html"><a href="t-test.html#two-sample-t-test"><i class="fa fa-check"></i><b>41.2</b> Two-Sample T-test</a><ul>
<li class="chapter" data-level="41.2.1" data-path="t-test.html"><a href="t-test.html#t-statistic-1"><i class="fa fa-check"></i><b>41.2.1</b> T-Statistic</a></li>
<li class="chapter" data-level="41.2.2" data-path="t-test.html"><a href="t-test.html#definitions-and-terminology-1"><i class="fa fa-check"></i><b>41.2.2</b> Definitions and Terminology</a></li>
<li class="chapter" data-level="41.2.3" data-path="t-test.html"><a href="t-test.html#hypotheses-2"><i class="fa fa-check"></i><b>41.2.3</b> Hypotheses</a></li>
<li class="chapter" data-level="41.2.4" data-path="t-test.html"><a href="t-test.html#decision-rule-2"><i class="fa fa-check"></i><b>41.2.4</b> Decision Rule</a></li>
</ul></li>
<li class="chapter" data-level="41.3" data-path="t-test.html"><a href="t-test.html#one-sided-test"><i class="fa fa-check"></i><b>41.3</b> One Sided Test</a><ul>
<li class="chapter" data-level="41.3.1" data-path="t-test.html"><a href="t-test.html#power-2"><i class="fa fa-check"></i><b>41.3.1</b> Power</a></li>
<li class="chapter" data-level="41.3.2" data-path="t-test.html"><a href="t-test.html#confidence-interval-1"><i class="fa fa-check"></i><b>41.3.2</b> Confidence Interval</a></li>
</ul></li>
<li class="chapter" data-level="41.4" data-path="t-test.html"><a href="t-test.html#references-5"><i class="fa fa-check"></i><b>41.4</b> References</a></li>
</ul></li>
<li class="chapter" data-level="42" data-path="uniform-distribution.html"><a href="uniform-distribution.html"><i class="fa fa-check"></i><b>42</b> Uniform Distribution</a><ul>
<li class="chapter" data-level="42.1" data-path="uniform-distribution.html"><a href="uniform-distribution.html#probability-density-function"><i class="fa fa-check"></i><b>42.1</b> Probability Density Function</a></li>
<li class="chapter" data-level="42.2" data-path="uniform-distribution.html"><a href="uniform-distribution.html#cumulative-density-function"><i class="fa fa-check"></i><b>42.2</b> Cumulative Density Function</a></li>
<li class="chapter" data-level="42.3" data-path="uniform-distribution.html"><a href="uniform-distribution.html#expected-values-12"><i class="fa fa-check"></i><b>42.3</b> Expected Values</a></li>
<li class="chapter" data-level="42.4" data-path="uniform-distribution.html"><a href="uniform-distribution.html#moment-generating-function-9"><i class="fa fa-check"></i><b>42.4</b> Moment Generating Function</a></li>
<li class="chapter" data-level="42.5" data-path="uniform-distribution.html"><a href="uniform-distribution.html#theorems-for-the-uniform-distribution"><i class="fa fa-check"></i><b>42.5</b> Theorems for the Uniform Distribution</a></li>
<li class="chapter" data-level="42.6" data-path="uniform-distribution.html"><a href="uniform-distribution.html#validity-of-the-distribution-8"><i class="fa fa-check"></i><b>42.6</b> Validity of the Distribution</a></li>
</ul></li>
<li class="chapter" data-level="43" data-path="variance-parameter.html"><a href="variance-parameter.html"><i class="fa fa-check"></i><b>43</b> Variance Parameter</a><ul>
<li class="chapter" data-level="43.1" data-path="variance-parameter.html"><a href="variance-parameter.html#defining-variance-with-expected-values"><i class="fa fa-check"></i><b>43.1</b> Defining Variance With Expected Values</a></li>
<li class="chapter" data-level="43.2" data-path="variance-parameter.html"><a href="variance-parameter.html#unbiased-estimator"><i class="fa fa-check"></i><b>43.2</b> Unbiased Estimator</a></li>
<li class="chapter" data-level="43.3" data-path="variance-parameter.html"><a href="variance-parameter.html#computational-formulae-2"><i class="fa fa-check"></i><b>43.3</b> Computational Formulae</a><ul>
<li class="chapter" data-level="43.3.1" data-path="variance-parameter.html"><a href="variance-parameter.html#computational-formula-population-variance"><i class="fa fa-check"></i><b>43.3.1</b> Computational Formula for <span class="math inline">\(\sigma\)</span>^2</a></li>
<li class="chapter" data-level="43.3.2" data-path="variance-parameter.html"><a href="variance-parameter.html#computational-formula-sample-variance"><i class="fa fa-check"></i><b>43.3.2</b> Computational Formula for <span class="math inline">\(s\)</span>^2</a></li>
<li class="chapter" data-level="43.3.3" data-path="variance-parameter.html"><a href="variance-parameter.html#corollary-alternative-computational-formula-for-s2"><i class="fa fa-check"></i><b>43.3.3</b> Corollary: Alternative Computational Formula for <span class="math inline">\(s^2\)</span></a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="44" data-path="weibull-distribution.html"><a href="weibull-distribution.html"><i class="fa fa-check"></i><b>44</b> Weibull Distribution</a><ul>
<li class="chapter" data-level="44.1" data-path="weibull-distribution.html"><a href="weibull-distribution.html#probability-distribution-function-4"><i class="fa fa-check"></i><b>44.1</b> Probability Distribution Function</a></li>
<li class="chapter" data-level="44.2" data-path="weibull-distribution.html"><a href="weibull-distribution.html#cumulative-distribution-function-7"><i class="fa fa-check"></i><b>44.2</b> Cumulative Distribution Function</a></li>
<li class="chapter" data-level="44.3" data-path="weibull-distribution.html"><a href="weibull-distribution.html#expected-values-13"><i class="fa fa-check"></i><b>44.3</b> Expected Values</a></li>
<li class="chapter" data-level="44.4" data-path="weibull-distribution.html"><a href="weibull-distribution.html#theorems-for-the-weibull-distribution"><i class="fa fa-check"></i><b>44.4</b> Theorems for the Weibull Distribution</a><ul>
<li class="chapter" data-level="44.4.1" data-path="weibull-distribution.html"><a href="weibull-distribution.html#validity-of-the-distribution-9"><i class="fa fa-check"></i><b>44.4.1</b> Validity of the Distribution</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="45" data-path="z-test-of-sample-proportions.html"><a href="z-test-of-sample-proportions.html"><i class="fa fa-check"></i><b>45</b> Z-test of Sample Proportions</a><ul>
<li class="chapter" data-level="45.1" data-path="z-test-of-sample-proportions.html"><a href="z-test-of-sample-proportions.html#one-sample-z-test"><i class="fa fa-check"></i><b>45.1</b> One-Sample Z-test</a><ul>
<li class="chapter" data-level="45.1.1" data-path="z-test-of-sample-proportions.html"><a href="z-test-of-sample-proportions.html#z-statistic"><i class="fa fa-check"></i><b>45.1.1</b> Z-Statistic</a></li>
<li class="chapter" data-level="45.1.2" data-path="z-test-of-sample-proportions.html"><a href="z-test-of-sample-proportions.html#definitions-and-terminology-2"><i class="fa fa-check"></i><b>45.1.2</b> Definitions and Terminology</a></li>
<li class="chapter" data-level="45.1.3" data-path="z-test-of-sample-proportions.html"><a href="z-test-of-sample-proportions.html#hypotheses-3"><i class="fa fa-check"></i><b>45.1.3</b> Hypotheses</a></li>
<li class="chapter" data-level="45.1.4" data-path="z-test-of-sample-proportions.html"><a href="z-test-of-sample-proportions.html#decision-rule-3"><i class="fa fa-check"></i><b>45.1.4</b> Decision Rule</a></li>
<li class="chapter" data-level="45.1.5" data-path="z-test-of-sample-proportions.html"><a href="z-test-of-sample-proportions.html#power-3"><i class="fa fa-check"></i><b>45.1.5</b> Power</a></li>
<li class="chapter" data-level="45.1.6" data-path="z-test-of-sample-proportions.html"><a href="z-test-of-sample-proportions.html#confidence-interval-2"><i class="fa fa-check"></i><b>45.1.6</b> Confidence Interval</a></li>
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<li class="chapter" data-level="45.2" data-path="z-test-of-sample-proportions.html"><a href="z-test-of-sample-proportions.html#references-6"><i class="fa fa-check"></i><b>45.2</b> References</a></li>
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<li class="chapter" data-level="46" data-path="z-test.html"><a href="z-test.html"><i class="fa fa-check"></i><b>46</b> Z-test</a><ul>
<li class="chapter" data-level="46.1" data-path="z-test.html"><a href="z-test.html#one-sample-z-test-1"><i class="fa fa-check"></i><b>46.1</b> One-Sample Z-test</a><ul>
<li class="chapter" data-level="46.1.1" data-path="z-test.html"><a href="z-test.html#z-statistic-1"><i class="fa fa-check"></i><b>46.1.1</b> Z-Statistic</a></li>
<li class="chapter" data-level="46.1.2" data-path="z-test.html"><a href="z-test.html#definitions-and-terminology-3"><i class="fa fa-check"></i><b>46.1.2</b> Definitions and Terminology</a></li>
<li class="chapter" data-level="46.1.3" data-path="z-test.html"><a href="z-test.html#hypotheses-4"><i class="fa fa-check"></i><b>46.1.3</b> Hypotheses</a></li>
<li class="chapter" data-level="46.1.4" data-path="z-test.html"><a href="z-test.html#decision-rule-4"><i class="fa fa-check"></i><b>46.1.4</b> Decision Rule</a></li>
<li class="chapter" data-level="46.1.5" data-path="z-test.html"><a href="z-test.html#power-4"><i class="fa fa-check"></i><b>46.1.5</b> Power</a></li>
<li class="chapter" data-level="46.1.6" data-path="z-test.html"><a href="z-test.html#confidence-interval-3"><i class="fa fa-check"></i><b>46.1.6</b> Confidence Interval</a></li>
</ul></li>
<li class="chapter" data-level="46.2" data-path="z-test.html"><a href="z-test.html#two-sample-t-test-1"><i class="fa fa-check"></i><b>46.2</b> Two-Sample T-test</a><ul>
<li class="chapter" data-level="46.2.1" data-path="z-test.html"><a href="z-test.html#t-statistic-2"><i class="fa fa-check"></i><b>46.2.1</b> T-Statistic</a></li>
<li class="chapter" data-level="46.2.2" data-path="z-test.html"><a href="z-test.html#definitions-and-terminology-4"><i class="fa fa-check"></i><b>46.2.2</b> Definitions and Terminology</a></li>
<li class="chapter" data-level="46.2.3" data-path="z-test.html"><a href="z-test.html#hypotheses-5"><i class="fa fa-check"></i><b>46.2.3</b> Hypotheses</a></li>
<li class="chapter" data-level="46.2.4" data-path="z-test.html"><a href="z-test.html#decision-rule-5"><i class="fa fa-check"></i><b>46.2.4</b> Decision Rule</a></li>
</ul></li>
<li class="chapter" data-level="46.3" data-path="z-test.html"><a href="z-test.html#one-sided-test-1"><i class="fa fa-check"></i><b>46.3</b> One Sided Test</a><ul>
<li class="chapter" data-level="46.3.1" data-path="z-test.html"><a href="z-test.html#power-5"><i class="fa fa-check"></i><b>46.3.1</b> Power</a></li>
<li class="chapter" data-level="46.3.2" data-path="z-test.html"><a href="z-test.html#confidence-interval-4"><i class="fa fa-check"></i><b>46.3.2</b> Confidence Interval</a></li>
</ul></li>
<li class="chapter" data-level="46.4" data-path="z-test.html"><a href="z-test.html#references-7"><i class="fa fa-check"></i><b>46.4</b> References</a></li>
</ul></li>
<li class="chapter" data-level="47" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html"><i class="fa fa-check"></i><b>47</b> Z-test of Sample Means</a><ul>
<li class="chapter" data-level="47.1" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#one-sample-z-test-2"><i class="fa fa-check"></i><b>47.1</b> One-Sample Z-test</a><ul>
<li class="chapter" data-level="47.1.1" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#z-statistic-2"><i class="fa fa-check"></i><b>47.1.1</b> Z-Statistic</a></li>
<li class="chapter" data-level="47.1.2" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#definitions-and-terminology-5"><i class="fa fa-check"></i><b>47.1.2</b> Definitions and Terminology</a></li>
<li class="chapter" data-level="47.1.3" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#hypotheses-6"><i class="fa fa-check"></i><b>47.1.3</b> Hypotheses</a></li>
<li class="chapter" data-level="47.1.4" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#decision-rule-6"><i class="fa fa-check"></i><b>47.1.4</b> Decision Rule</a></li>
<li class="chapter" data-level="47.1.5" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#power-6"><i class="fa fa-check"></i><b>47.1.5</b> Power</a></li>
<li class="chapter" data-level="47.1.6" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#confidence-interval-5"><i class="fa fa-check"></i><b>47.1.6</b> Confidence Interval</a></li>
</ul></li>
<li class="chapter" data-level="47.2" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#two-sample-z-test"><i class="fa fa-check"></i><b>47.2</b> Two-Sample Z-test</a><ul>
<li class="chapter" data-level="47.2.1" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#z-statistic-3"><i class="fa fa-check"></i><b>47.2.1</b> Z-Statistic</a></li>
<li class="chapter" data-level="47.2.2" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#definitions-and-terminology-6"><i class="fa fa-check"></i><b>47.2.2</b> Definitions and Terminology</a></li>
<li class="chapter" data-level="47.2.3" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#hypotheses-7"><i class="fa fa-check"></i><b>47.2.3</b> Hypotheses</a></li>
<li class="chapter" data-level="47.2.4" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#decision-rule-7"><i class="fa fa-check"></i><b>47.2.4</b> Decision Rule</a></li>
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<li class="chapter" data-level="47.3" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#one-sided-test-2"><i class="fa fa-check"></i><b>47.3</b> One Sided Test</a><ul>
<li class="chapter" data-level="47.3.1" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#power-7"><i class="fa fa-check"></i><b>47.3.1</b> Power</a></li>
<li class="chapter" data-level="47.3.2" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#confidence-interval-6"><i class="fa fa-check"></i><b>47.3.2</b> Confidence Interval</a></li>
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<li class="chapter" data-level="47.4" data-path="z-test-of-sample-means.html"><a href="z-test-of-sample-means.html#references-8"><i class="fa fa-check"></i><b>47.4</b> References</a></li>
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<li class="chapter" data-level="48" data-path="references-9.html"><a href="references-9.html"><i class="fa fa-check"></i><b>48</b> References</a></li>
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            <section class="normal" id="section-">
<div id="gamma-function" class="section level1">
<h1><span class="header-section-number">17</span> Gamma Function</h1>
<p>The Gamma Function is a function used frequently in statistical theory. It has properties that permit simplified calculations and is used in defining many probability distributions, particularly within the Exponential family of distributions.</p>
<div id="gamma-function-definition" class="section level2">
<h2><span class="header-section-number">17.1</span> Definition</h2>
<p><span class="math display">\[\Gamma(x) = \int\limits_0^{\infty}t^{x-1}e^{-t}dt\]</span></p>
<p>Note that the definition defines a function of <span class="math inline">\(x\)</span> that is integrated over <span class="math inline">\(t\)</span>. Thus, for each value of <span class="math inline">\(x\)</span>, a curve is defined, and the Gamma function calculates the area under the curve defined by <span class="math inline">\(x\)</span>.</p>
</div>
<div id="theorems-for-the-gamma-function" class="section level2">
<h2><span class="header-section-number">17.2</span> Theorems for the Gamma Function</h2>
<div id="gamma-function-reduction-lemma" class="section level3">
<h3><span class="header-section-number">17.2.1</span> Lemma</h3>
<p><span class="math display">\[\left[ -t^{x-1} e^{-t} \right]_{t = 0}^{t = \infty} = 0\]</span></p>
<p><em>Proof:</em></p>
<p><span class="math display">\[\begin{aligned}
\left[ -t^{x-1} e^{-t} \right]_{t = 0}^{t = \infty}
 &amp;= \lim_{t\to\infty}\big(-t^{x - 1} e^{-t}\big) -
       0^{x - 1} e^{-0} \\
 &amp;= \lim_{t\to\infty}\big(-t^{x - 1} e^{-t}\big) - 0 \\
 &amp;= \lim_{t\to\infty}\big(-t^{x - 1} e^{-t}\big) \\
 &amp;= - \lim_{t\to\infty}\frac{t^{x - 1}}{e^t} \\
 &amp;= - \lim_{t\to\infty} \left\{exp\left[ (x - 1) \ln t - t\right] \right\} \\
 &amp;= - \lim_{t\to\infty} \left\{ exp \left[(x - 1) \cdot t \cdot \Big( \frac{\ln t}{t} - 1\Big) \right]\right\} \\
^{[1]} &amp;= \lim_{t\to\infty} \left\{ exp \left[(x - 1) \cdot t \cdot \Big( 0 - 1\Big) \right]\right\} \\
&amp;= \lim_{t\to\infty} \left\{ exp \left[- (x - 1) \cdot t \right]\right\} \\
 &amp;= \lim_{t\to\infty} \frac{1}{e^{(x - 1) \cdot t}} \\
 &amp;= 0
\end{aligned}\]</span></p>
<blockquote>
<ol style="list-style-type: decimal">
<li>L’H^{o}pital’s Rule: <span class="math inline">\(\lim_{x\to u}\frac{f(x)}{g(x)} = \lim_{x\to u} \frac{f&#39;(x)}{g&#39;(x)}\)</span>. This implies <span class="math inline">\(\lim_{x\to \infty} \frac{\ln x}{x} = \lim_{x\to \infty} \frac{\frac{1}{x}}{1} = \lim_{x \to \infty} \frac{1}{x} = 0\)</span></li>
</ol>
</blockquote>
</div>
<div id="gamma-function-reduction-relation" class="section level3">
<h3><span class="header-section-number">17.2.2</span> Theorem: The Reduction Relation</h3>
<p><span class="math inline">\(\Gamma(x) = (x-1) \cdot \Gamma(x)\)</span></p>
<p><em>Proof:</em></p>
<p>The proof relies on integration by parts. Let:</p>
<p><span class="math display">\[\begin{aligned}
u &amp;= t^{x-1} \\
du &amp;= (x - 1) \cdot t^{(x-2)} \\\\
v &amp;= -e^{-t} \\
dv &amp;= e^{-t} dt
\end{aligned}\]</span></p>
<p>Integration by parts yields:</p>
<p><span class="math display">\[\begin{aligned}
\Gamma(x) &amp;= \int\limits_0^{\infty}t^{x-1}e^{-t}dt \\
    &amp;= u \cdot v - \int\limits_0^\infty v \cdot du \\
    &amp;= \Big[t^{x-1} \cdot - e^{-t}\Big]_{t=0}^{t=\infty} -
        \int\limits_0^\infty-e^{-t} \cdot (x-1)\cdot t^{(x-2)} dt \\
    &amp;= -\Big[t^{x-1} \cdot e^{-t}\Big]_{t=0}^{t=\infty} -
        (- (x - 1)) \int\limits_0^\infty e^{-t} \cdot t^{(x-2)} dt \\
^{[1]}    &amp;= -0 +
        (x - 1) \int\limits_0^\infty e^{-t} \cdot t^{(x-2)} dt \\
    &amp;= (x - 1) \int\limits_0^\infty e^{-t} \cdot t^{((x-1)-1)} dt \\
    &amp;= (x - 1) \int\limits_0^\infty t^{((x-1)-1)} \cdot e^{-t} dt \\
    &amp;= (x - 1) \cdot \Gamma(x - 1)
\end{aligned}\]</span></p>
<blockquote>
<ol style="list-style-type: decimal">
<li><a href="gamma-function.html#gamma-function-reduction-lemma">17.2.1</a></li>
</ol>
</blockquote>
</div>
<div id="corollary" class="section level3">
<h3><span class="header-section-number">17.2.3</span> Corollary</h3>
<p><span class="math display">\[\Gamma(x) = \frac{1}{x} \cdot \Gamma(x + 1)\]</span></p>
<p><em>Proof:</em></p>
<p>Theorem <a href="gamma-function.html#gamma-function-reduction-relation">17.2.2</a> establishes</p>
<p><span class="math display">\[ \Gamma(x) = (x - 1) \cdot \Gamma(x - 1)\]</span></p>
<p>Let <span class="math inline">\(y = x + 1\)</span>. Then</p>
<p><span class="math display">\[\begin{aligned}
\Gamma(y) &amp;= (y - 1) \cdot \Gamma(y - 1) \\
\Rightarrow \Gamma(x + 1)
    &amp;= (x + 1 - 1) \cdot \Gamma(x + 1 - 1) \\
    &amp;= x \cdot \Gamma(x) \\
\Rightarrow \frac{1}{x} \cdot \Gamma(x + 1) &amp;= \Gamma(x) \\
\Rightarrow \Gamma(x) &amp;= \frac{1}{x} \cdot \Gamma(x + 1)
\end{aligned}\]</span></p>
<p>This allows the recurrence relation to move toward <span class="math inline">\(\Gamma(0)\)</span> for any value of <span class="math inline">\(x\)</span>. Note, however, that <span class="math inline">\(\Gamma(0)\)</span> is undefined. Thus, solutions for the Gamma Function may be determined for positive integers, since <span class="math inline">\(\Gamma(1)\)</span> can be solved. On the other hand, <span class="math inline">\(\Gamma(-1)\)</span> can not be solved, and the recurrence relation results in a zero denominator. Hence, the Gamma Function is defined for all <span class="math inline">\(x \in \mathbb{R}\)</span> so long as <span class="math inline">\(x \not\in \mathbb{Z^-}\)</span></p>
</div>
<div id="theorem-3" class="section level3">
<h3><span class="header-section-number">17.2.4</span> Theorem:</h3>
<p><span class="math display">\[\Gamma\Big(\frac{1}{2}\Big) = \sqrt{\pi}\]</span></p>
<p><em>Proof:</em></p>
<p><span class="math display">\[\begin{aligned}
\Gamma\Big(\frac{1}{2}\Big)
  &amp;= \int\limits_0^\infty t^{-\frac{1}{2}} e ^{-t} dt \\
  &amp;= \frac{2}{2} \int\limits_0^\infty t^{-\frac{1}{2}} e ^{-t} dt \\
  &amp;= 2 \int\limits_0^\infty \frac{1}{2} \cdot t ^{-\frac{1}{2}}
      e^{-(\sqrt{t})^2} dt \\
  &amp;= 2 \int\limits_0^\infty \frac{1}{2} \cdot t ^{-\frac{1}{2}}
      e^{-(\sqrt{t})^2} dt \\
^{[1]}  &amp;= 2 \int\limits_0^\infty \frac{1}{2} \cdot x ^{-1}
      e^{-x^2} \cdot 2\cdot x \ dx \\
  &amp;= 2 \int\limits_0^\infty \frac{2x}{2x} e^{-x^2} dx \\
  &amp;= 2 \int\limits_0^\infty e^{-x^2} dx \\
^{[2]}  &amp;= 2 \cdot \frac{\sqrt{\pi}}{2} \\
  &amp;= \sqrt{\pi}
\end{aligned}\]</span></p>
<blockquote>
<ol style="list-style-type: decimal">
<li><span class="math inline">\(x = \sqrt{t}\)</span>; <span class="math inline">\(t = x^2\)</span>; and <span class="math inline">\(dt = 2x dx\)</span></li>
<li>Theorem <a href="gaussian-integral.html#gaussian-integral-sqrt-pi">18.1.2</a></li>
</ol>
</blockquote>
</div>
<div id="theorem-4" class="section level3">
<h3><span class="header-section-number">17.2.5</span> Theorem:</h3>
<p>Let <span class="math inline">\(c\)</span> be a constant such that <span class="math inline">\(c &gt; 0\)</span>. Then</p>
<p><span class="math display">\[
  \int_0^\infty t^x e^{-ct} \cdot dt = \frac{\Gamma(x+1)}{c^{x+1}}
\]</span></p>
<p><em>Proof:</em></p>
<p><span class="math display">\[\begin{aligned}
\int\limits_0^\infty t^x e^{-ct} dt \\
^{[1]}  &amp;= \int\limits_0^\infty \big(\frac{y}{c}\big) ^ x e ^{-c \frac{y}{c}} \frac{dy}{c} \\
  &amp;= \frac{1}{c^x} \cdot \frac{1}{c}
      \int\limits_0^\infty y ^ x e ^{-y} dy \\
&amp;= \frac{1}{c^{x + 1}}
      \int\limits_0^\infty y ^ {(x+1)-1} e ^{-y} dy \\
^{[2]}  &amp;= \frac{1}{c^{x + 1}} \cdot \Gamma(x + 1) \\
  &amp;= \frac{\Gamma(x + 1)}{c^{x + 1}}
\end{aligned}\]</span></p>
<blockquote>
<ol style="list-style-type: decimal">
<li>Let <span class="math inline">\(y = ct\)</span>. Then <span class="math inline">\(y = \frac{y}{c}\)</span> and <span class="math inline">\(dt = \frac{dy}{c}\)</span>.</li>
<li><span class="math inline">\(\int\limits_0^\infty y^{(x+1)-1} e^{-y} dy\)</span> satisfies the form of a Gamma function. <a href="gamma-function.html#gamma-function-definition">17.1</a></li>
</ol>
</blockquote>
</div>
</div>
<div id="references-1" class="section level2">
<h2><span class="header-section-number">17.3</span> References</h2>
<ul>
<li>Pennsylvania State University, Elberly College of Science, STAT 414/415, <a href="https://onlinecourses.science.psu.edu/stat414/node/142" class="uri">https://onlinecourses.science.psu.edu/stat414/node/142</a></li>
<li>Theodore Hatch Whitfield, Lecture Notes, E156 <em>Mathematical Foundations of Statistical Software</em>.</li>
</ul>

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