Page 152 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 145

are all less than z is

Since there are n ways of choosing the variables to be maximum, we have

When n = 2, Eq. (4.98) reduces to

which is the same as Eq. (4.86) (Prob. 4.25) with fx(z) = fy(z) = f (2) and FX(z) = FY(z) = F(z).

4.31. Let XI, . . . , X, be n independent r.v.'s each with the identical pdf f (x). Let W = min(Xl, . . . , X,).
Find the pdf of W.
The probability P(w < W < w + dw) is equal to the probability that one of the r.v.3 falls in (w, w + dw)
and all others are greater than w. The probability that one of Xi (i = 1, . . . , n) falls in (w, w + dw) and all
others are greater than w is

Since there are n ways of choosing the variables to be minimum, we have

When n = 2, Eq. (4.1 00) reduces to

which is the same as Eq. (4.88) (Prob. 4.26) with f,(w) = f,(w) =f (w) and F,(w) = F,(w) = F(w).

Let Xi, i = 1, . . . , n, be n independent gamma r.v.'s with respective parameters (ai, A), i = 1, . . . ,
n. Let

Show that Y is also a gamma r.v. with parameters (C=, ai, 1).
We prove this proposition by induction. Let us assume that the proposition is true for n = k; that is,

k
is a gamma r.v. with parameters (8, A) = ( C ai, A).
i= 1
k+ 1
Let W=Z+Xk+l = 2 Xi
i= 1
Then, by the result of Prob. 4.20, we see that W is a gamma r.v. with parameters (/3 + a,+ ,, A) =
(Elf ,' a,, 1). Hence, the proposition is true for n = k + 1. Next, by the result of Prob. 4.20, the proposition
is true for n = 2. Thus, we conclude that the proposition is true for any n 2 2.