Download MTH 7241: Fall 2009: Prof. C. King Assignment 7 Due date

MTH 7241: Fall 2009: Prof. C. King
Assignment 7
Due date: Wednesday, December 9.
Problems:
1). Suppose that a certain component has a lifetime which is an exponential
random variable with mean 100 hours. A machine uses 20 of these components – assume that their lifetimes are independent.
a) If exactly 4 components fail within the first 50 hours, what is the expected
time until another component fails?
Answer: 100/16
b) If exactly 4 components fail within the first 50 hours, what is the probability that no more components will fail within the next 10 hours?
Answer: e−1.6
2). Let X be an exponential random variable with rate λ. Determine
E[X|X < c] by using the identity
E[X] = E[X|X < c] P (X < c) + E[X|X > c] P (X > c)
Answer: [1 − (1 + λc)e−λc ]/λ(1 − e−λc )
3). Let X1 , X2 , X3 be independent exponential r.v.’s with rates λ1 , λ2 , λ3
respectively. Calculate
P (X1 < X2 < X3 )
Answer: λ2 λ1 /(λ2 + λ3 )(λ1 + λ2 + λ3 )
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4). Consider a post office with two clerks. Three people A,B,C enter simultaneously. A and B go directly to clerks, while C waits until a clerk is free
before he begins service. Assume the three service times are independent
exponential with rate λ. What is the probability that A is still in the post
office after both B and C have left?
Answer: 1/4
5). Let X and Y be independent exponential r.v.’s with rates λ and µ
respectively. Let c > 0. Show that the conditional density function of X
conditioned on X + Y = c is
fX|X+Y (x|c) =
(λ − µ)e−(λ−µ)x
1 − e−(λ−µ)c
0<x<c
Use this result to compute E[X|X + Y = c].
Answer: 1/(λ − µ) − c e−(λ−µ)c /(1 − e−(λ−µ)c )
6). Customers arrive at a bank at a Poisson rate λ. Suppose two customers
arrived during the first hour. What is the probability that (a) both arrived
during the first 20 minutes?, (b) at least one arrived during the first 20
minutes?
Answer: 1/9, 5/9
7). The number of hours between successive train arrivals at a station is
uniformly distributed on [0, 1]. Passengers arrive according to a Poisson
process with rate 7 per hour. Suppose a train has just left the station. Let
X denote the number of people that get on the next train. Find E[X] and
V AR[X].
Answer: E[X] = λ/2, V AR[X] = λ2 /12 + λ/2
8). N1 (t) and N2 (t) are independent Poisson processes, with rates 2 and 4
respectively. Starting at an arbitrary time, compute the probability that at
least two arrivals from N1 occur before three arrivals from N2 .
Answer: µ2 (µ2 + 4λµ + 6λ2 )/(µ + λ)4
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9). Events occur according to a Poisson process with rate λ = 2 per hour.
(a) What is the probability that no event occurs between 8pm and 9pm?
Answer: e−λ
(b) Starting at noon, what is the expected time at which the fourth event
occurs?
Answer: 4/λ
(c) What is the probability that two or more events occur between 6pm and
8pm?
Answer: 1 − e−2λ − 2λe−2λ
10). Suppose that people arrive at a bus stop in accordance with a Poission
process with rate λ. The bus departs at time t. Let X denote the total
amount of waiting time of all those that get on the bus at time t. We want
to determine V AR[X]. Let N (t) denote the number of arrivals by time t.
(a) Find E[X|N (t)].
Answer: N (t) t/2
(b) Show that V AR[X|N (t)] = N (t) t2 /12.
(c) Find V AR[X].
Answer: λ t3 /3
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