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title: 隨機變數 (Random Variable)
description:
published: true
date: 2025-12-28T19:20:55.599Z
date: 2025-12-28T19:24:32.966Z
tags:
editor: ckeditor
dateCreated: 2025-12-28T16:41:51.938Z
-->
<h2 id="discrete" class="toc-header"> Discrete</h2>
<h3 id="probability-mass-function-pmf" class="toc-header"> Probability Mass Function (PMF)</h3>
<h2>Discrete</h2>
<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><strong>0 ≤ p<sub>i</sub> ≤ 1 and ∑ p<sub>i</sub> = 1</strong></p>
<p>Probability<br>
<svg style="vertical-align: -0.312ex;" xmlns="http://www.w3.org/2000/svg" width="1.76ex" height="1.751ex" role="img" viewBox="0 -636 778 774" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1-TEX-N-2264" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"></g></g></g></svg></p>
<p>Expectation</p>
<h3 id="cumulative-distribution-function-cdf" class="toc-header"> Cumulative Distribution Function (CDF)</h3>
<p></p>
<h2 id="continuous" class="toc-header"> Continuous</h2>
<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>
<p><br><strong>Probability</strong></p>
<figure class="image"><img src="data:image/png;base64,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"></figure>
<p><br><strong>Expectation</strong></p>
<figure class="image"><img src="data:image/png;base64,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"></figure>
<p>&nbsp;</p>
<h3>Cumulative Distribution Function (CDF)</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>Expectation</p>
<h3 id="cumulative-distribution-function-cdf-1" class="toc-header"> Cumulative Distribution Function (CDF)</h3>
<p></p>
<h3 id="symmetric" class="toc-header"> Symmetric</h3>
<h3>Cumulative Distribution Function (CDF)</h3>
<p>&nbsp;</p>
<h3>Symmetric</h3>
<p>If there is a point that,</p>
<p>Then,</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>
<p>The middle value of the random variable.<br>
For median, set p to 0.5.</p>
<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>
<h3>Median and Quantiles</h3>
<p>The middle value of the random variable.<br>For median, set p to 0.5.</p>
<h3>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>