Chapter 6: Q.6.48 (page 274)
If are independent and identically distributed exponential random variables with the parameter , compute
(a) role="math" localid="1647168400394" ;
(b) role="math" localid="1647168413468"
Short Answer
(a)
(b)
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Chapter 6: Q.6.48 (page 274)
If are independent and identically distributed exponential random variables with the parameter , compute
(a) role="math" localid="1647168400394" ;
(b) role="math" localid="1647168413468"
(a)
(b)
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If are independent exponential random variables with respective parameters and , find the distribution of . Also compute .
Suppose that n points are independently chosen at random on the circumference of a circle, and we want the probability that they all lie in some semicircle. That is, we want the probability that there is a line passing through the center of the circle such that all the points are on one side of that line, as shown in the following diagram:

Let P1, ... ,Pn denote the n points. Let A denote the event that all the points are contained in some semicircle, and let Ai be the event that all the points lie in the semicircle beginning at the point Pi and going clockwise for 180â—¦, i = 1, ... , n.
(a) Express A in terms of the Ai.
(b) Are the Ai mutually exclusive?
(c) Find P(A).
The number of people who enter a drugstore in a given hour is a Poisson random variable with parameter λ = 10. Compute the conditional probability that at most 3 men entered the drugstore, given that 10 women entered in that hour. What assumptions have you made?
The time that it takes to service a car is an exponential random variable with rate .
(a) If A. J. brings his car in at timeand M. J. brings her car in at time t, what is the probability that M. J.’s car is ready before A. J.’s car? (Assume that service times are independent and service begins upon arrival of the car.)
(b) If both cars are brought in at time 0, with work starting on M. J.’s car only when A. J.’s car has been completely serviced, what is the probability that M. J.’s car is ready before time ?
Let W be a gamma random variable with parameters (t, β), and suppose that conditional on W = w, X1, X2, ... , Xn are independent exponential random variables with rate w. Show that the conditional distribution of W given that X1 = x1, X2 = x2, ... , Xn = xn is gamma with parameters t + n, β + n i=1 xi .
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