Introduction to Probability Models 10th. Edition by Sheldon M. Ross | 9780123756862 | 0123756863 | Holooly

Statistical Mechanics

Introduction to Probability Models

85 SOLVED PROBLEMS

Question: 3.27

In an election, candidate A receives n votes, and candidate B receives m votes where n > m. Assuming that all orderings are equally likely, show that the probability that A is always ahead in the count of votes is (n − m)/(n + m). ...

Verified Answer:

Let

P_{n,m}

denote the desired prob...

Question: 3.22

An insurance company supposes that the number of accidents that each of its policyholders will have in a year is Poisson distributed, with the mean of the Poisson depending on the policyholder. If the Poisson mean of a randomly chosen policyholder has a gamma distribution with density function ...

Verified Answer:

Let X denote the number of accidents that a random...

Question: 2.27

(Variance of the Normal Random Variable) Let X be normally distributed with parameters μ and σ². Find Var(X). ...

Verified Answer:

Recalling (see Example 2.22) that E[X] = μ, we hav...

Question: 3.19

(The Variance of a Compound Random Variable) Let X1, X2, … be independent and identically distributed random variables with distribution F having mean μ and variance σ², and assume that they are independent of the nonnegative integer valued random variable N. As noted in Example 3.11, where its ...

Verified Answer:

Whereas we could obtain

E[S^{2}]

by...

Question: 3.18

Independent trials, each resulting in a success with probability p, are performed in sequence. Let N be the trial number of the first success. Find Var(N). ...

Verified Answer:

Let Y = 1 if the first trial results in a success,...

Question: 3.15

Independent trials, each of which is a success with probability p, are performed until there are k consecutive successes. What is the mean number of necessary trials? ...

Verified Answer:

Let

N_{k}

denote the number of nece...

Question: 3.21

Suppose that X and Y are independent continuous random variables having densities fX and fY, respectively. Compute P{X < Y}. ...

Verified Answer:

Conditioning on the value of Y yields

P\{X\...

Question: 3.17

In the match problem of Example 2.31 involving n, n > 1, individuals, find the conditional expected number of matches given that the first person did not have a match. ...

Verified Answer:

Let X denote the number of matches, and let [latex...

Question: 3.13

A miner is trapped in a mine containing three doors. The first door leads to a tunnel that takes him to safety after two hours of travel. The second door leads to a tunnel that returns him to the mine after three hours of travel. The third door leads to a tunnel that returns him to his mine after ...

Verified Answer:

Let X denote the time until the miner reaches safe...

Question: 3.9

Let X1 and X2 be independent exponential random variables with rates μ1 and μ2. Find the conditional density of X1 given that X1 + X2 = t. ...

Verified Answer:

To begin, let us first note that if f(x, y) is the...

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