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# Homer and Marge enter a coee shop simultaneouslyHomer to get a plain coee and Marge an espresso.

Homer and Marge enter a coee shop simultaneously—Homer to get a plain coee and Marge

an espresso. At this coee shop, the amount of time X1 it takes to get a coee 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 coee 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 coee?

(c) Now, dene Y = min(X1 , X2 ) as the minimum of X1 and X2 , that is, the amount of time

it takes until whoever is served rst. We wish to nd the probability distribution of Y by the

distribution function technique.

(i) First nd P (Y > y).

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(ii) Using the result in part i, nd the cdf of Y .

(ii) What is the pdf of Y ?

(ii) Find the mean and the variance of Y .