Variance of the time of gambler’s ruin. Let 1; 2; : : : be independent with P. i D 1/ D p and P. i D 1/ D q D 1  p where p < 1=2. Let Sn D S0 C 1 C  C n. In Example 4.3 we showed that if V0 D minfn  0 W Sn D 0g then ExV0 D x=.12p/. 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 D 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?