A Joint Distribution Function:

Let S be the sample space of a random experiment. Let X and Y be two r.v.'s defined on S. Then the pair ( X, Y) is called a two-dimensional r.v. if each of X and Y associates a real number with every element of S. The joint cumulative distribution function (or joint cdf) of X and Y, denoted by F _Xy( x, y), is the function defined by

Two r.v.'s X and Y will be called independent if

for every values of x and y.

B Marginal Distribution Function:

Since { X ? ?} and { Y ? ?}

so that

The cdf's F _X( x) and F _Y( y), when obtained by Eqs. (6.41a) and (6.41b), are referred to as the marginal cdf's of X and Y, respectively.

C Joint Probability Mass Functions:

Let ( X, Y) be a discrete two-dimensional r.v. and ( X, Y) takes on the values ( x _i, y _j) for a certain allowable set of integers i and j. Let

The function p _XY( x _i, y _j) is called the joint probability mass function (joint pmf) of ( X, Y).

Properties of f _XY( x,y):

The joint cdf of a discrete two-dimensional r.v.