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# System prepended metadata

title: 機率TP7
tags: [Probability]

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---
title: 機率TP7
tags: Probability
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# Discrete Random Variables：<br>Joint PMFs, Conditioning and Independence

## context

## Total Probability Theorem
![](https://i.imgur.com/MpkOgJG.png)

## Conditioning a Random Variable on Another
- Let $X$ and $Y$ be two random variables associated with the same experiment. The conditional PMF $p_{X|Y}$ if $X$ given $Y$ is defined as  
![](https://i.imgur.com/reWOe5Z.png)
![](https://i.imgur.com/QeZfroU.png)
- The conditional PMF can also be used to calculate the marginal PMFs
    - $p_X(x) = \sum_yp_{X, Y}(x,y) = \sum_y p_Y(y)p_{X|Y}(x|y)$
- Visualization if the conditional PMF $p_{X|Y}$
    - ![](https://i.imgur.com/LEVZ5iy.png)
