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Unformatted text preview: .3 we showed that if V0 = min{n 0 : Sn = 0}
then Ex V0 = x/(1 2p). The aim of this problem is to compute the variance
of V0 . (a) Show that (Sn (p q )n)2 n(1 (p q )2 ) is a martingale. (b) Use
this to conclude that when S0 = x the variance of V0 is
x· 1 (p q )2
(p q )3 (c) Why must the answer in (b) be of the form cx?
5.12. Generating function of the time of gambler’s ruin. Continue with the
setup of the previous problem. (a) Use the exponential martingale and our
stopping theorem to conclude that if ✓ 0, then e✓x = Ex ( (✓) V0 ). (b) Let
0 < s < 1. Solve the equation (✓) = 1/s, then use (a) to conclude
!x
p
1
1 4pqs2
V0
Ex (s ) =
2ps
(c) Why must the answer in (b) be of the form f (s)x ?
5.13. Consider a favorable game in which the payo↵s are 1, 1, or 2 with probability 1/3 each. Use the results of Example 5.12 to compute the probability
we ever go broke (i.e, our winnings Wn reach $0) when we start with $i. 177 5.6. EXERCISES 5.14. A branching process can be turned into a random walk i...
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 Spring '10
 DURRETT
 The Land

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