Mathematics 375 – Probability and Statistics 1 Midterm Exam 2a November 16, 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 shipment of 14 television sets, 5 have defective “on” switches and will not power up. Out of the shipment, 7 sets are selected at random. What is the probability that at most 3 will not power up?

II. A) (10) The number of defects Y in any 1-foot segment of a magnetic tape is a Poisson

random variable with mean 5. What is the probability that a 1-foot segment contains exactly 4 defects?

B) (15) Show, by any applicable method, that if number of defects in one 1-foot segment is Y_{1} and the number in a second 1-foot segment (not overlapping the first) is Y_{2}, then the total number of defects, Y = Y_{1} + Y_{2}, has a Poisson distribution with mean 10. (Assume defects in the two segments are independent.)

III. 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 = 3 and β = 2. 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 29 weeks. C) (15) If 6 exposed animals are chosen at random, use your answer from part B to

estimate the probability at least 2 of them survive longer than 29 weeks. D) (15) The cost of treating an exposed animal is C = 5 + .2Y^{2}. Find the mean and

variance of C.

IV. A) (10) The lifetime of a certain type of small motor is uniformly distributed on the

interval [0, 30]. What is the probability that the motor will last more than 10 years? B) (10) Now, assume the lifetime of the motors is normally distributed with mean 15

years and standard deviation 4 years. The manufacturer replaces free all motors 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 3% of the motors that fail. What should the length of the warrantee be?

Extra Credit (10) Let Z be a standard normal random variable. What is the momentgenerating function for Y = 8Z − 4.3? What is the distribution of Y?