48. Give another proof of the result of Example 3.17 by computing the moment generating function of...
Question:
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.
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