Page 151 - Schaum's Outlines - Probability, Random Variables And Random Processes

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

FUNCTIONS OF N RANDOM VARIABLES
4.29. Let X, Y, and Z be independent standard normal r.v.'s. Let W = (X2 + Y2 + Z2)lI2. Find the
pdf of W.
We have

and F,(W)= P(W I w) = P(X2 + Y2 + z2 < w2)

where Rw = ((x, y, z): x2 + y2 + z2 I w2). Using spherical coordinates (Fig. 4-10), we have

dx dy dz = r2 sin 8 dr d8 dq
e-r2/2 r 2 sin 8 dr dB dq
and

- sin 8 d8 [e-.'12r2 dr
- -& 5,'= dm

Thus, the pdf of W is

Fig. 4-10 Spherical coordinates.

4.30. Let XI, . . . , X, be n independent r.v.'s each with the identical pdf f (x). Let Z = max(X,, . . . , X,).
Find the pdf of 2.
The probability P(z < Z < z + dz) is equal to the probability that one of the r.v.'s falls in (z, z + dz)
and all others are less than z. The probability that one of Xi (i = 1, . . . , n) falls in (z, z + dz) and all others