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title: 隨機變數 (Random Variable) title: 隨機變數 (Random Variable)
description: description:
published: true published: true
date: 2025-12-28T19:20:55.599Z date: 2025-12-28T19:24:32.966Z
tags: tags:
editor: ckeditor editor: ckeditor
dateCreated: 2025-12-28T16:41:51.938Z dateCreated: 2025-12-28T16:41:51.938Z
--> -->
<h2 id="discrete" class="toc-header"> Discrete</h2> <h2>Discrete</h2>
<h3 id="probability-mass-function-pmf" class="toc-header"> Probability Mass Function (PMF)</h3> <h3>Probability Mass Function (PMF)</h3>
<p>A set of probability value <strong>p<sub>i</sub></strong> assigned to each of the values taken by the discrete random variables <strong>x<sub>i</sub></strong>.</p> <p>A set of probability value <strong>p<sub>i</sub></strong> assigned to each of the values taken by the discrete random variables <strong>x<sub>i</sub></strong>.</p>
<p><strong>0 ≤ p<sub>i</sub> ≤ 1 and ∑ p<sub>i</sub> = 1</strong></p> <figure class="image"><img src="data:image/png;base64,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"></figure>
<p>Probability<br> <p><br><strong>Probability</strong></p>
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<p>Expectation</p> <p><br><strong>Expectation</strong></p>
<h3 id="cumulative-distribution-function-cdf" class="toc-header"> Cumulative Distribution Function (CDF)</h3> <figure class="image"><img src="data:image/png;base64,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"></figure>
<p></p> <p>&nbsp;</p>
<h2 id="continuous" class="toc-header"> Continuous</h2> <h3>Cumulative Distribution Function (CDF)</h3>
<h3 id="probability-density-function-pdf" class="toc-header"> Probability Density Function (PDF)</h3> <figure class="image"><img src="data:image/png;base64,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"></figure>
<figure class="image"><img src="data:image/png;base64,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"></figure>
<p>&nbsp;</p>
<h2>Continuous</h2>
<h3>Probability Density Function (PDF)</h3>
<p>Probabilistic properties of a continuous random variable.</p> <p>Probabilistic properties of a continuous random variable.</p>
<p>Expectation</p> <p>Expectation</p>
<h3 id="cumulative-distribution-function-cdf-1" class="toc-header"> Cumulative Distribution Function (CDF)</h3> <h3>Cumulative Distribution Function (CDF)</h3>
<p></p> <p>&nbsp;</p>
<h3 id="symmetric" class="toc-header"> Symmetric</h3> <h3>Symmetric</h3>
<p>If there is a point that,</p> <p>If there is a point that,</p>
<p>Then,</p> <p>Then,</p>
<p>Is the expectation of this random variable, equal to the point of symmetry.</p> <p>Is the expectation of this random variable, equal to the point of symmetry.</p>
<h3 id="median-and-quantiles" class="toc-header"> Median and Quantiles</h3> <h3>Median and Quantiles</h3>
<p>The middle value of the random variable.<br> <p>The middle value of the random variable.<br>For median, set p to 0.5.</p>
For median, set p to 0.5.</p> <h3>Variance</h3>
<h3 id="variance" class="toc-header"> Variance</h3> <p>A positive quantity that measures the spread of the distribution of the random variable about its mean value.<br>Larger values of the variance indicate that the distribution is more spread out.</p>
<p>A positive quantity that measures the spread of the distribution of the random variable about its mean value.<br>
Larger values of the variance indicate that the distribution is more spread out.</p>