- Suppose 2 random variables X and Y
- Suppose X and Y can have categorical values from 1 to 6
- You need to consider
probability distribution of random variable X,
probability distribution of random variable Y,
probability distribution of random variable pair of X,Y
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Probability distribution of random variables X and Y can be expressed by probability mass functions
$$$P_X(x)$$$, $$$P_Y(y)$$$
$$$P_X(x)$$$: probability mass function of random variable X
$$$P_Y(x)$$$: probability mass function of random variable Y
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$$$P_X(x=1) or P_X(1)$$$: means probability of 1 occuring from random variable X
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Joint probability mass function of X and Y: $$$P_{XY}(x,y)$$$
$$$P_{XY}(x=2,y=3)$$$: means
- probability of event {x=2,y=3} occuring
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Marginal probability mass function
It's probability distribution function wrt one random variable from joint probability mass function
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How to calculate marginal probability mass function
$$$P_X(x)=\sum\limits_{y_i} P_{XY}(x,y_i)$$$
$$$P_Y(y)=\sum\limits_{x_i} P_{XY}(x_i,y)$$$
$$$P_X(A) \\
= P_{XY}(A,A) + P_{XY}(A,B) + P_{XY}(A,C) + P_{XY}(A,D) + P_{XY}(A,E) + P_{XY}(A,F) \\
= 0.02$$$
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Conditional probability mass function
When y is already occurred and fixed,
$$$P_{X \mid Y}(x \mid y) = \dfrac{P_{XY}(x, y)}{P_{Y}(y)}$$$
When x is already occurred and fixed,
$$$P_{Y \mid X}(y \mid x) = \dfrac{P_{XY}(x, y)}{P_{X}(x)}$$$