48. Give another proof of the result of Example 3.17 by computing the moment generating function of N i=1Xi and then differentiating to obtain its moments.

Hint: Let

φ(t) = E

, exp t



N i=1 Xi

-

= E

, E

, exp t



N i=1 Xi

!

!

!

!

!

N

--

Now, E

, exp t



N i=1 Xi

!

!

!

!

!

N = n

-

= E

, exp t

n i=1 Xi

- = (φX(t))n since N is independent of the Xs where φX(t) = E[etX] is the moment generating function for the Xs. Therefore,

φ(t) = E

"

(φX(t))N #

Differentiation yields

φ

(t) = E

"

N(φX(t))N−1φ

X(t)#

,

φ(t) = E

"

N(N − 1)(φX(t))N−2(φ

X(t))2 + N(φX(t))N−1φ

X(t)#

Evaluate at t = 0 to get the desired result.

49. A and B play a series of games with A winning each game with probability p. The overall winner is the first player to have won two more games than the other.

(a) Find the probability that A is the overall winner.

(b) Find the expected number of games played.