Page 64 - Probability, Random Variables and Random Processes

RANDOM VARIABLES [CHAP 2

(b) FindP(X> l)ifn= 6andp=0.1.

(a) Recall that the binomial expansion formula is given by

Thus, by Eq. (2.36),

(b) NOW P(X > 1) = 1 - P(X = 0) - P(X = 1)

2.17. Let X be a Poisson r.v. with parameter A.

(a) Show that p,(x) given by Eq. (2.40) satisfies Eq. (2.1 7).
(b) Find P(X > 2) with 1 = 4.

(h) With A = 4, we have

and

Thus,

CONTINUOUS RANDOM VARIABLES AND PDF'S
2.18. Verify Eq. (2.1 9).

From Eqs. (1.27) and (2.10), we have

for any E 2 0. As Fx(x) is continuous, the right-hand side of the above expression approaches 0 as E + 0.
Thus, P(X = x) = 0.

2.19. The pdf .of a continuous r.v. X is given by
3 O<x<l

0 otherwise
Find the corresponding cdf FX(x) and sketch fx(x) and F,(x).