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phidias committed 2025-12-28 19:40:01 +00:00
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title: 隨機變數 (Random Variable)
description:
published: true
date: 2025-12-28T19:38:38.698Z
date: 2025-12-28T19:40:00.675Z
tags:
editor: ckeditor
dateCreated: 2025-12-28T16:41:51.938Z
@@ -30,8 +30,20 @@ dateCreated: 2025-12-28T16:41:51.938Z
<tbody>
<tr>
<td style="text-align:center;width:300px;"><strong>Distribution Function</strong></td>
<td style="text-align:center;width:300px;"><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Discrete</strong></span></td>
<td style="text-align:center;width:300px;"><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Continuous</strong></span></td>
<td style="text-align:center;width:500px;"><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Discrete</strong></span></td>
<td style="text-align:center;width:500px;"><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Continuous</strong></span></td>
</tr>
<tr>
<td style="text-align:center;"><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Cumulative</strong></span><br><br><span style="font-family:Arial, Helvetica, sans-serif;">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>.</span></td>
<td>
<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>
</td>
<td>
<figure class="image"><img 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truncated
<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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 truncated
</td>
</tr>
<tr>
<td style="text-align:center;"><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Jointly Distributed</strong></span></td>
@@ -87,11 +99,6 @@ dateCreated: 2025-12-28T16:41:51.938Z
<p><br><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Expectation</strong></span></p>
<figure class="image"><img src="data:image/png;base64,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"></figure>
<p>&nbsp;</p>
<h3><span style="font-family:Arial, Helvetica, sans-serif;">Cumulative Distribution Function (CDF)</span></h3>
<p><span style="font-family:Arial, Helvetica, sans-serif;">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>.</span></p>
<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><span style="font-family:Arial, Helvetica, sans-serif;">Continuous</span></h2>
<h3><span style="font-family:Arial, Helvetica, sans-serif;">Probability Density Function (PDF)</span></h3>
<p><span style="font-family:Arial, Helvetica, sans-serif;">Probabilistic properties of a continuous random variable.</span></p>
@@ -100,10 +107,3 @@ dateCreated: 2025-12-28T16:41:51.938Z
<p><span style="font-family:Arial, Helvetica, sans-serif;"><strong>Expectation</strong></span></p>
<figure class="image"><img src="data:image/png;base64,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"></figure>
<p>&nbsp;</p>
<h3><span style="font-family:Arial, Helvetica, sans-serif;">Cumulative Distribution Function (CDF)</span></h3>
<p><span style="font-family:Arial, Helvetica, sans-serif;">Probabilistic properties of a continuous random variable.</span></p>
<figure class="image"><img src="data:image/png;base64,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 truncated
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<figure class="image"><img 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truncated
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