Page 71 - Probability, Random Variables and Random Processes

CHAP. 21 RANDOM VARIABLES

By Eqs. (2.26) and (2.36), and letting q = 1 - p, we have

n n !
= Ck pkqn
k=O (n - k)! k! -

Letting i = k - 1 and using Eq. (2.72), we obtain

Next,

n
- n! k n-k
- lk(k- - k)!k!
k = 0

Similarly, letting i = k - 2 and using Eq. (2.72), we obtain
n-2 (n - 2)!
E[X(X - I)] = n(n - l)p2 C ~~q~-~-~
i=o (n - 2 - i)! i!

= n(n - l)p2(p + q)"-2 = n(n - l)p2
Thus, E(X2) = E[X(X - l)] + E(X) = n(n - l)p2 + np
and by Eq. (2.31),
ax2 = Var(x) = n(n - l)p2 + np - (n~)~ = np(1 - p)

2.29. Let X be a Poisson r.v. with parameter 1. Verify Eqs. (2.42) and (2.43).
By Eqs. (2.26) and (2.40),

Next,

Thus,