Problem 71E

Let Y denote a geometric random variable with probability of success p.

a Show that for a positive integer a,

P ( Y > a ) = qa .

b Show that for positive integers a and b,

P ( Y > a + b|Y > a) = qb = P(Y > b).

This result implies that, for example, P(Y > 7|Y > 2) = P(Y > 5). Why do you think this property is called the memoryless property of the geometric distribution?

c In the development of the distribution of the geometric random variable, we assumed that the experiment consisted of conducting identical and independent trials until the first success was observed. In light of these assumptions, why is the result in part (b) “obvious”?

Solution:

Step 1 of 4:

Let Y denote a geometric random variable with probability of success P.

We have to show that

- For a positive integer a,

P(Y>a) = .

(b) For positive integers a and b,

P(Y>a+b/Y>a) = P(Y>b).

(c) Why is the result in part (b) is obvious.