Solved: Suppose that X and Y are independent, identically

QUESTION:

Suppose that X and Y are independent, identically distributed, geometric random variables with parameter p. Show that P{X = i 1 X + Y = n) = 1 _, n -1 i = 1, . .. , n - 1. Solution. Consider repeatedly and independently tossing a coin with probability of heads p. We can interpret P{X = i 1 X + Y = n) as the probability that we obtained a head for the first time on the ith toss given that we obtained a head for the second time on the nth toss. We can then argue, intuitively, that given that the second head occurred on the nth toss, the first head is equally likely to have come up at any toss between 1 and n - 1. To establish this precisely, note that we have P{X - '1 X Y _ ) _ P{X = i, X + Y = n) _ P{X = i)P{Y = n - i) - 1. + -n - - . P{X + Y = n) P{X + Y = n) Also and It follows that P{X = i) = p(1 _ p)i-l , P{Y = n - i) = p{l _ p)n-i-l , for i 2: 1, for n - i 2: 1. P{X = i)P{Y = n -i) = p p , { 2 {1 _ ) n-2 0, if i = 1, . . . , n - 1, otherwise. Therefore, for any i and j in the range [1, n - 1] , we have P{X = i 1 X + Y = n) = P{X = j 1 X + Y = n) . Hence P{X=i IX+Y = n) = 1 ' n - i=l, ... , n - 1.