Homer and Marge enter a coffee shop simultaneously -- Homer to get a plain coffee and Marge an espresso. At this coffee shop, the amount of time X1 it takes to get a coffee is exponentially distributed with mean 2 minutes, and the amount of time X2 it takes to receive an espresso is exponentially distributed with mean 10 minutes at the coffee shop. Suppose that Homer and Marge are immediately served, and the service times X1 and X2 are independent.
(a) Find the joint probability density function of X1 and X2.
(b) What is the probability that Marge will get her espresso before Homer gets his coffee?
(c) Now, define Y = min(X1,X2) as the minimum of X1 and X2, that is, the amount of time it takes until whoever is served first. We wish to find the probability distribution of Y by the distribution function technique.
(i) First find P(Y > y).
(ii) Using the result in part (i), find the cdf, pdf, mean and variance of Y .