Joint probability distribution

Joint probability distribution

If you have a pair of random variables $X$ and $Y$ the joint probability distribution function, $F_{XY}(x,y)$ tells you the following probability. $$ F_{XY}(x,y) = P( X \le x \wedge Y \le y ) $$ This function does not appear that often and it is more common to work with the joint probability mass function if the random variables are discrete: $$ f_{XY}(x,y) = P( X=x \wedge Y=y ) $$ You can calculate the marginal distribution, $f_X(x)$, for the random variable $X$ from the joint probability mass function using: $$ f_X(x) = P( X=x ) = \sum_{i=0}^\infty f_{XY}(x,y_i) $$ Similarly the marginal distribution, $f_Y(y)$, for the random variable $Y$ can be found using: $$ f_Y(y) = P( Y=y ) = \sum_{i=0}^\infty f_{XY}(x_i,y) $$

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