- 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 ================================================================================ 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 ================================================================================ $$$P_X(x=1) or P_X(1)$$$: means probability of 1 occuring from random variable X ================================================================================ 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 ================================================================================ Marginal probability mass function It's probability distribution function wrt one random variable from joint probability mass function ================================================================================ How to calculate marginal probability mass function $$$P_X(x)=\sum\limits_{y_i} P_{XY}(x,y_i)$$$ $$$P_Y(y)=\sum\limits_{x_i} P_{XY}(x_i,y)$$$ $$$P_X(A) \\ = P_{XY}(A,A) + P_{XY}(A,B) + P_{XY}(A,C) + P_{XY}(A,D) + P_{XY}(A,E) + P_{XY}(A,F) \\ = 0.02$$$ ================================================================================ 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, $$$P_{Y \mid X}(y \mid x) = \dfrac{P_{XY}(x, y)}{P_{X}(x)}$$$