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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:
- \(\mathbf{E} (N)\)
- \(\mathbf{E} (D)\)
- \(\mathbf{E} (T)\)
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 \(\mathrm{Poisson}(\mu)\).
- What is the distribution of \(X_n\) ?
- 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.
- What is the chance that a particular square inch is not hit by any drops during a given 10-second period ?
- 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?
- 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
- \(\mathbf{P}(X+Y = 4) \)
- \(\mathbf{E}[(X+Y)^2]\)
- \(\mathbf{P}(X+Y + Z= 4) \)
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.
- What is the expected number of pages with no mistakes ? What is the variance of the number of pages with no mistakes ?
- 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 ?
- What, approximately, is the probability that the book has two or more pages with mistakes ?
[Pitman p235, #15]
Looking for a rare couple
A certain advertising firm needs to find a recently married couple who are both born on April 19th. They send the intern to the city hall in NYC to look through the records.
- If she looks through 50,000 records, what is the chance that she finds at least one such couple?
- If you were to approximate the above probability with a limit theorem what distribution would you use? Explain why.
- If she relaxes her search criteria to only look for couples with the same birthday, what is the chance she finds at least one such couple after looking through 50,000 records?
- If you were to approximate the above probability with a limit theorem what distribution would you use? Explain why.
- Before starting, the intern is interested in estimating how long this task might take. How many records must she consider to have a 80% chance of finding a couple matching the first search criteria? What about the second search criteria?
In the above, assume that the date people are born is uniform over the year and each person’s birthday is independent of each other person’s birthday. Also, assume that the day one is born does not influence one’s choice of spouse.
