Question

A) Find the probability that two independent geometric random variables W1 and W2 with parameters p1 and p2 are equal, i.e., P(W1 = W2). You can express this as P(W1 = W2) = ∑(from k=1 to [infinity]) P(W1 = k, W2 = k).

b) Calculate the probability that W1 is less than W2, i.e., P(W1 < W2). Express this in terms of k (similar to Part a) and use the fact that if X follows a geometric distribution with parameter p, then P(X > n) = (1 - p)^n for every n ≥ 1.

c) Determine the probability that W1 is greater than W2, i.e., P(W1 > W2).

Answer 1

To find the probability that two independent geometric random variables, W1 and W2, are equal, we calculate the sum of their equal values. To calculate the probability that W1 is less than W2, we subtract the probability of W1 and W2 being equal from 1. To find the probability that W1 is greater than W2, we subtract the probability of W1 being less than or equal to W2 from 1.

Step-by-step explanation:

To find the probability that two independent geometric random variables, W1 and W2, are equal, we need to calculate the sum of the event where both variables have the same value, P(W1=k, W2=k), for all possible values of k.

For part b, to calculate the probability that W1 is less than W2, we use the fact that P(X > n) = (1 - p)^n. So, P(W1 < W2) = 1 - P(W1 = W2) = 1 minus the probability that W1 and W2 are equal.

For part c, we can simply subtract the probability of W1 being less than or equal to W2 from 1 to find the probability that W1 is greater than W2.