Question

Suppose X and Y are independent Geometric random variables, with E(X)=4 and E(Y)=3/2.

a. Find the probability that X and Y are equal, i.e., find P(X=Y).

b. Find the probability that X is strictly larger than Y, i.e., find P(X>Y). [Hint: Unlike Problem 1b, we do not have symmetry between X and Y here, so you must calculate.]

Answer 1

a) We are given here that E(X) = 4 and E(Y) = 3/2, therefore the probability of getting a success on any trial for these 2 cases here is computed as: 1/4 = 0.25 and 2/3

The required probability here is computed as:

Applying the sum of an infinite GP, we get here: ( first term as (1/4)(2/3) and common ratio as (3/4)(1/3))

Therefore (2/9) = 0.2222 is the required probability here.

b) Here, there is no symmetry so we need to do it manually as:

Again this becomes an infinite GP with first term as (2/3)(3/4) = 0.5 and common ratio as (1/3)(3/4) = 0.25

Therefore the infinite sum of the GP here is computed as:

Therefore (2/3) = 0.6667 is the required probability here.