Question

A dealer sells A, B, and C. Suppose the time until the next A is sold has an exponential distribution with a mean of 5 days, the time until the next pair of B is sold has an exponential distribution with a mean of 8 days, and the time until the next C is sold has an exponential distribution with a mean of 20 days. Assume that these three times are independent.

(a) What is the probability that next item that the dealer sells will be sold between 3 and 6 days from now?

(b) What is the probability that the next item sold by the dealer will be a A?

Answer 1

a)

minimum of exponential follow exponential distribution with sum of rates

1/mean = 1/5 + 1/8 + 1/20 = 0.375

hence

mean = 2.66666

P(3 <X < 6)

= e^(-3/ 2.66666 ) - e^(-6/ 2.66666 )

= 0.219252

b)

probability that the next item sold by the dealer will be a A

= lambda_A / (lambda_A + lambda_B + lambda_C)

= 1/5 / (1/5 + 1/8 +1/20)

= 0.53333