For 22. In part (c), only show that the geometric random variable is memoryless;

Question

For 22. In part (c), only show that the geometric random variable is memoryless; showing that a distribution on the positive integers is necessarily memoryless is harder.

22. Let X be a geometric random variable with parameter p, and n and m be nonnegative integers.

(a) For what values of n is P (X = n) maximum?

(b) What is the probability that X is even?

(c) Show that the geometric is the only distribution on the positive integers with the memoryless property: P (X > n + m | X > m) = P (X > n)

I just need the first question answered. Thanks.

Explanation / Answer

Question 22

(a) Here P(X = n) = (1-p)n-1p

so for maximum value of probability dP/dn = 0

dP/dn = (n-1) (1-p)n-2

so at n = 1 we have the maximum value of P( x = n) which would be equal to p

(b) Here Pr(x = Even) = P(x = 2) + P(X = 4) + ....

= (1-p)p + (1-p)3p+ ...

= (1-p)p [1 + (1-p)2 + ...]

= (1-p)p [1/ (1 -(1-p)2)]

= p(1-p) * 1/ [1 - (1 + p2-2p)]

= p(1-p)/(2p - p2)

= (1-p)/(2-p)

(c) Here we have to prove that

P(X > n + m l X > m)

Here P(x > m) = 1 - P(x   m) = 1 - [1 - (1-p)m] = (1-p)m

P(x > n + m l x > m) = [1 - (1-p)n+m]/ (1-p)m = (1-p)n

which is equals to P(X > n)

so we can say that geometric distribution has memoryless propoerty.

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