# Tag Archives: JCM_math230_HW7_S15

## Handing back tests

A professor randomly hands back test in a class of $$n$$ people paying no attention to the names on the paper. Let $$N$$ denote the number of people who got the right test. Let $$D$$ denote the pairs of people who got each others tests. Let $$T$$ denote the number of groups of three who none got the right test but yet among the three of them that have each others tests. Find:

1. $$\mathbf{E} (N)$$
2. $$\mathbf{E} (D)$$
3. $$\mathbf{E} (T)$$

## Up by two

Suppose two teams play a series of  games, each producing a winner and a loser, until one time has won two more games than the other. Let $$G$$ be the number of games played until this happens. Assuming your favorite team wins each game with probability $$p$$, independently of the results of all previous games, find:

1. $$P(G=n)$$ for $$n=2,3,\dots$$
2. $$\mathbf{E}(G)$$
3. $$\mathrm{Var}(G)$$

[Pittman p220, #18]

## Population

A population contains $$X_n$$ individuals  at time $$n=0,1,2,\dots$$ . Suppose that $$X_0$$ is distributed as $$\mathrm{Poisson}(\mu)$$. Between time $$n$$ and $$n+1$$ each of the $$X_n$$ individuals dies with probability $$p$$ independent of the others. The population at time $$n+1$$ is comprised of the survivors together with a random number of new immigrants who arrive independently in numbers distributed according to $$\mathrm{Poisson}(\mu)$$.

1. What is the distribution of $$X_n$$ ?
2. What happens to this distribution as $$n \rightarrow \infty$$ ? Your answer should depended on $$p$$ and $$\mu$$. In particular, what is $$\mathbf{E} X_n$$ as $$n \rightarrow \infty$$ ?

[Pittman [236, #18]

## Raindrops are falling

Raindrops are falling at an average rate of 30 drops per square inch per minute.

1. What is the chance that a particular square inch is not hit by any drops during a given 10-second period ?
2. If one draws a circle of radius 2 inches on the ground, what is the chance that 4 or more drops hits inside the circle over a two-minute period?
3. If each drop is a big drop with probability 2/3 and a small drop with probability 1/3, independent of the other drops, what is the chance that during 10 seconds a particular square inch gets hit by precisely four big drops and five small ones?

[Pitman p. 236, #17, Modified by Mattingly]

## Mixing Poisson Random Variables 1

Assume that  $$X$$, $$Y$$, and $$Z$$ are independent Poisson random variables, each with mean 1. Find

1. $$\mathbf{P}(X+Y = 4)$$
2. $$\mathbf{E}[(X+Y)^2]$$
3. $$\mathbf{P}(X+Y + Z= 4)$$

## Random Errors in a Book

A book has 200 pages. The number of mistakes on each page is a Poisson random variable with mean 0.01, and is independent of the number of mistakes on all other pages.

1. What is the expected number of pages with no mistakes ? What is the variance of the number of pages with no mistakes ?
2. A person proofreading the book finds a given mistake with probability 0.9 . What is the expected number of pages where this person will find a mistake ?
3. What, approximately, is the probability that the book has two or more pages with mistakes ?

[Pitman p235, #15]