Page 153 - Schaum's Outlines - Probability, Random Variables And Random Processes

FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS [CHAP. 4

Let XI, . . . , X, be n independent exponential r.v.'s each with parameter A. Let

Show that Y is a gamma r.v. with parameters (n, A).
We note that an exponential r.v. with parameter 1 is a gamma r.v. with parameters (1, 1) (Prob. 2.24).
Thus, from the result of Prob. 4.32 and setting ai = 1, we conclude that Y is a gamma r.v. with parameters
(n, 4.

Let Z,, . . . , Z, be n independent standard normal r.v.'s. Let

i= 1
Find the pdf of Y.
Let Y, = Zi2. Then by Eq. (4.75) (Prob. 4.7), the pdf of is

Now, using Eq. (2.80), we can rewrite

and we recognize the above as the pdf of a gamma r.v. with parameters (4, 4) [Eq. (2.76)]. Thus, by the
result of Prob. 4.32, we conclude that Y is the gamma r.v. with parameters (n/2, 3) and

When n is an even integer, T(n/2) = [(n/2) - I]!, whereas when n is odd, T(n/2) can be obtained from
T(a) = (a - l)r(a - 1) [Eq. (2.78)] and r(4) = fi [Eq. (2.80)].
Note that Equation (4.102) is referred to as the chi-square (x2) density function with n degrees of
freedom, and Y is known as the chi-square (X2) r.v. with n degrees of freedom. It is important to recognize
that the sum of the squares of n independent standard normal r.v.3 is a chi-square r.v. with n degrees of
freedom. The chi-square distribution plays an important role in statistical analysis.

Let XI, X, , and X, be independent standard normal r.v.'s. Let
Y, = XI + X, + X3
Y2 = X, - X,
Y, = X2 - X3
Determine the joint pdf of Y,, Y2 , and Y3 .
Let y, = x, + x, + x3
Y2 = Xl - x2
Y3 = x2 - x3
By Eq. (4.32), the jacobian of transformation (4.103) is