Homer and Marge enter a co ffee shop simultaneously -- Homer to get a plain co ffee and Marge an espresso. At this co ffee shop, the amount of time X1 it takes to get a co ffee 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 co ffee 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 co ffee?
(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 fi rst. We wish to fi nd 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 .