Page 93 - Schaum's Outlines - Probability, Random Variables And Random Processes

86 MULTIPLE RANDOM VARIABLES [CHAP 3

which can be reduced to

The conditional mean of X, given that Y = y, and the conditional variance of X, given that Y = y,
are given by similar expressions. Note that the conditional mean of Y, given that X = x, is a function
of x alone. Similarly, the conditional mean of X, given that Y = y, is a function of y alone (Prob.
3.40).

3.9 N-VARIATE RANDOM VARIABLES
In previous sections, the extension from one r.v. to two r.v.'s has been made. The concepts can be
extended easily to any number of r.v.'s defined on the same sample space. In this section we briefly
describe some of the extensions.

A. Definitions:
Given an experiment, the n-tuple of r.v.'s (XI, X,, . . . , X,) is called an n-variate r.v. (or n-
dimensional random vector) if each Xi, i = 1, 2, . . . , n, associates a real number with every sample
point E S. Thus, an n-variate r.v. is simply a rule associating an n-tuple of real numbers with every
y E S.
Let (XI, . . . , X,) be an n-variate r.v. on S. Then its joint cdf is defined as

Note that

The marginal joint cdf's are obtained by setting the appropriate Xi's to +GO in Eq. (3.61). For
example,

A discrete n-variate r.v. will be described by a joint pmf defined by
pxl ... xn(~l, . . . , x,) = P(X = x . . . , X, = x,) (3.65)
The probability of any n-dimensional event A is found by summing Eq. (3.65) over the points in the
n-dimensional range space RA corresponding to the event A :

Properties of p,, . . . (x, , . . . , x,,) :

The marginal pmf's of one or more of the r.v.'s are obtained by summing Eq. (3.65) appropriately.
For example,