###### Introduction to Probability Models

295 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: 3.18

## 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: 3.13

## 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 ...

Let X denote the time until the miner reaches safe...
Question: 7.EX.8

## A machine in use is replaced by a new machine either when it fails or when it reaches the age of T years. If the lifetimes of successive machines are independent with a common distribution F having density f , show that (a) the long-run rate at which machines are replaced equals ...

(a) The number of replaced machines by time t cons...
Question: 3.11

## Suppose that the expected number of accidents per week at an industrial plant is four. Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of 2. Assume also that the ...

Letting N denote the number of accidents and [late...
Question: 2.45

## Calculate the distribution of X + Y when X and Y are independent Poisson random variables with means λ1 and λ2, respectively. ...

\phi_{X+Y}(t)=\phi_{X}(t)\,\phi_{Y}(t)[/lat...
Question: 11.EX.32

## Show that if X and Y have the same distribution then Var((X + Y)/2) ≤ Var(X) Hence, ...

\begin{aligned}\operatorname{Var}[(X+Y) / 2...
Question: 11.EX.23

## For a nonhomogeneous Poisson process with intensity function λ(t), t ≥ 0, where ∫0^∞ ...

Let m(t)=\int_{0}^{t} \lambda(s)\, d s[/lat...
Question: 11.EX.16

## Suppose n balls having weights w1, w2, . . . , wn are in an urn. These balls are ...

Let \begin{aligned}& c(\lambda)=\max _{...