###### 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). ...

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 ...

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). ...

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 ...

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). ...

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? ...

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}. ...

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. ...

Let X denote the number of matches, and let [latex...
Question: 3.13