Question
A pizza place has two phones. On each phone the
waiting time until the first call is exponentially
distributed with mean one minute. Each phone is
not influenced by the other. Let X be the shorter of
the two waiting times and let Y be the longer.
It can be shown that the joint pdf of X and Y is
f(x,y) = 2e-(a+), 0<x<y<x
a. Determine the marginal density of X.
b. Determine the conditional density of Y given
X = x.
c. Determine the probability that Y is greater
than 2, given that X = 1.
a. Are X and Y independent? Explain.
e. Determine the conditional mean of Y given
X = x. Is E(YX = x) a linear function of x?
f. Determine the conditional variance of Y given
X = x.
Expand
Transcribed Image Text

A pizza place has two phones. On each phone the waiting time until the first call is exponentially distributed with mean one minute. Each phone is not influenced by the other. Let X be the shorter of the two waiting times and let Y be the longer. It can be shown that the joint pdf of X and Y is f(x,y) = 2e-(a+), 0<x<y<x a. Determine the marginal density of X. b. Determine the conditional density of Y given X = x. c. Determine the probability that Y is greater than 2, given that X = 1. a. Are X and Y independent? Explain. e. Determine the conditional mean of Y given X = x. Is E(YX = x) a linear function of x? f. Determine the conditional variance of Y given X = x.

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