Question

Let x and y be independent random variables each geometrically distributed with parameter 0.8. What is the probability that both x and y are greater than 2?

a) 0.8
b) 0.16
c) 0.36
d) 0.64

Answer 1

The probability that both X and Y are greater than 2, if X and Y are independent and identically geometrically distributed random variables with parameter 0.8, is 0.0016.

Step-by-step explanation:

The question asks for the probability that both X and Y are greater than 2, given that X and Y are independent random variables each geometrically distributed with parameter 0.8. The geometric distribution gives the probability that the first occurrence of success requires k independent trials, each with success probability p. Specifically, the probability we want is P(X>2) multiplied by P(Y>2) because the two events are independent.

To find P(X>2), we use the fact that for a geometrically distributed random variable, P(X>k) = (1-p)k. We apply this with k=2 and p=0.8 to get P(X>2) = (1-0.8)2 = (0.2)2 = 0.04. Since X and Y are independent and identically distributed, P(Y>2) = P(X>2) = 0.04. Therefore, the probability that both X and Y are greater than 2 is P(X>2 and Y>2) = P(X>2) * P(Y>2) = 0.04 * 0.04 = 0.0016.