Let {Xn}n=1 be independent identically distributed random variables with values in {1,0,1,2,...}, and with mean (1,0). Define

and

n

Sn =1+?Xi, ,n1, S0 =1,

i=1

=min{n0:Sn =0}

(1) Show that this stopping time is finite with probability one: P(

1

(2) Derive the estimate: E( ) || .

When applying MCT or DCT, make sure not to skip steps

4. (3pts+3pts) Let {Xn}, be independent identically distributed random variables with values in {-1, 0, 1, 2, ...}, and with mean / E (-1,0). Define n Sn = 1+ S Xi, ,n 21, So = 1, i=1 and T = min {n > 0 : Sn = 0} (1) Show that this stopping time is finite with probability one: P(T