Joint Probability Distributions

If X and Y are discrete, this distribution can be described with a joint probability mass function. If X and Y are continuous, this distribution can be described with a joint probability density function.
If we are given a joint probability distribution for X and Y , we can obtain the individual probability distribution for X or for Y (and these are called the Marginal Probability Distributions.
Because the the probability mass functions for X and Y appear in the margins of the table i.e. column and row totals), they are often referred to as the Marginal Distributions for X and Y .
When there are two random variables of inter-
est, we also use the term bivariate probability distribution or bivariate distribution to refer to the joint distribution. Marginal Probability Mass Function
If X and Y are discrete random variables with joint probability mass function fXY (x, y)
When asked for E(X) or V (X) (i.e. val-
ues related to only 1 of the 2 variables) but you
are given a joint probability distribution, ?rst
calculate the marginal distribution fX(x) and
work it as we did before for the univariate case
(i.e. for a single random variable).