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Definition for two random variables

When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined. For example, for a pair of random variables X, Y {\displaystyle X,Y}, the joint CDF F X Y {\displaystyle F_{XY}} is given by

where the right-hand side represents the probability that the random variable X {\displaystyle X} takes on a value less than or equal to x {\displaystyle x} and that Y {\displaystyle Y} takes on a value less than or equal to y {\displaystyle y}.

Example of joint cumulative distribution function:

For two continuous variables X and Y: Pr (a < X < b and c < Y < d) = ∫ a b ∫ c d f (x, y) d y d x; {\displaystyle \Pr(a<X<b{\text{ and }}c<Y<d)=\int _{a}^{b}\int _{c}^{d}f(x,y)\,dy\,dx;}

For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range of X and Y, and here is the example:

given the joint probability mass function in tabular form, determine the joint cumulative distribution function.

Solution: using the given table of probabilities for each potential range of X and Y, the joint cumulative distribution function may be constructed in tabular form: