Mathematics 375 – Probability and Statistics 1 Midterm Exam 2b November 18, 2005

Directions

Do all work in the blue exam booklet. There are 100 regular points and 10 Extra Credit points. If you know general formulas for means, variances, etc. you can use them without comment (i.e. you don’t need to rederive them in you solution).

I. (10) In a group of 12 chickens, 5 are infected with the H5N1 “bird flu” virus and the others are not. If 7 chickens are selected randomly from the group, what is the probability 2 or more have the virus?

II. A) (15) Assume that Y_{1} has a Poisson distribution with mean λ_{1} and Y_{2} has Poisson

distribution with mean λ_{2}. Also assume that Y_{1} = y_{1} and Y_{2} = y_{2} are independent events for all y_{1} ≥ 0 and y_{2} ≥ 0. Show, by any applicable method, that Y = Y_{1} + Y_{2} also has a Poisson distribution, with mean λ = λ_{1} + λ_{2}.

B) (10) The number of defects Y in any 2-foot segment of a magnetic tape is a Poisson random variable with mean 3. Use part A to find the probability that a 4-foot segment contains exactly 8 defects, assuming the necessary independence. (You can use part A even if you didn’t completely see how to do that part.)

III. A) (10) The lifetime of a certain type of battery is uniformly distributed on the interval

[0, 25] (years). What is the probability that the battery will last more than 10 years? B) (10) Now, assume the lifetime of the batteries is normally distributed with mean 10

years and standard deviation 4 years. The manufacturer replaces free all batteries that fail while under warrantee. But to avoid going out of business because of the cost of doing the free replacements, the manufacturer must set the warrantee so that they will have to replace no more than 5% of the batteries that fail. What should the length of the warrantee be?

IV. In a biomedical research experiment, the survival time Y (in weeks) of an animal exposed to a certain fatal disease is a random variable with a gamma distribution with α = 4 and β = 3. A) (5) What is the mean survival time of a randomly selected exposed animal? B) (10) Use Tchebysheff’s Theorem to estimate the probability that a single exposed

animal will survive longer than 30 weeks. C) (15) If 6 exposed animals are chosen at random, use your answer from part B to

estimate the probability no more than 2 of them survive longer than 30 weeks. D) (15) The cost of treating an exposed animal is C = 4 + .25Y^{2}. Find the mean and

variance of C.

Extra Credit (10) Let Z be a standard normal random variable. What is the momentgenerating function for Y = 3Z + 5? What is the distribution of Y?