Mathematics 375 – Probability and Statistics 1 Final Examination – December 17, 2005

Directions: Do problems I - X in the blue exam booklet. There are 200 possible regular points and 20 Extra Credit points. Notes:

1) If an answer can be expressed in terms of factorials, binomial coefficients, the gamma function, etc., it is preferable to leave it in that form.

2) Even if you don’t see how to derive a complete solution to a problem, don’t leave it blank. Partial credit will be given for relevant definitions, ideas that could lead to solutions, and so forth.

I. 20 students in a statistics class were asked to report the number of pets in their families, giving the following data:

\begin{matrix}3, & 3, & 2, & 4, & 1, & 0, & 1, & 2, & 3, & 5, \\ 1, & 2, & 2, & 1, & 1, & 0, & 2, & 4, & 3, & 1 \end{matrix}

A) (10) Construct a relative frequency histogram for this data using one “bin” for each

integer. B) (10) How many data points are within one sample standard deviation of the sample

mean?

II. Suppose that you have a key ring with N keys, exactly one of which is your house key. You get home after dark and can’t see the keys or the ring. So you try the keys in your front door lock one by one until you find the right one. A) (5) Assuming that you are not careful and you mix up the keys before each try, what

is the probability that you first find the correct key on the yth try (y = 1, 2, …)? B) (5) Let Y_{A} be the discrete random variable giving the number of the trial on which

you find the correct key by the method from part A. What is the expected value of Y_{A}?

C) (5) Same question as part A, but now you put aside each key as you try it so it does

not get mixed back in with the ones you have not tried. D) (5) Let Y_{C} be the discrete random variable giving the number of the trial on which

you first find the correct key by the method from part C. What is the expected value of Y_{C}? Which method is superior?

III. A tea distributor randomly places a miniature porcelain figurine of one of 10 different endangered animals in each of its boxes of tea. A) (15) What is the probability of getting a complete set of all 10 animals if you purchase

exactly 10 boxes of the tea? Explain how you are deriving your answer. B) Extra Credit (10) What is the expected number of boxes of tea you need to purchase

to obtain a complete set of all 10 animals? (Hint: You get one in the first box you buy. What is the expected number of boxes until you get a second, different animal? After you have two, what is the expected number of boxes to get the third? etc.)

A) (5) State Bayes’ Rule. B) (15) According to the American Lung Association, 7% of the population has lung

disease. Of those people having lung disease, 90% are smokers. Of those not having lung disease, 74.7% are nonsmokers. What is the probability that a smoker has lung disease?

V. The number of customers arriving at a furniture store in a one-hour period has a Poisson distribution with mean 5. A) (10) What is the probability that more than 7 customers arrive in one hour? B) (10) What is the probability that more than 7 customers arrive in exactly 4 out of

the 8 hours that the store is open in one day? (Assume the numbers in each hour are independent.)

VI. The pdf for a continuous random variable Y has the form

f(y) = \{ \begin{matrix}k(9 − y^{2}) & if −3 ≤ y ≤ 3 \\ 0 & otherwise \end{matrix}

A) (10) Determine the value of k. B) (15) What are the expected value and variance of Y?

VII. Professional golfers’ tee shot distances are normally distributed with μ = 282.2 yards and σ = 8.12 yards. A) (10) What is the probability that a tee shot goes more than 300 yards? B) (10) What is the probability that a tee shot goes between 270 and 290 yards?

VIII. (15) Two phone calls are made independently in a fixed 1-hour period. Each call occurs at a random time uniformly distributed on the interval [0, 1]. Find the joint density function for the times Y_{1}, Y_{2} of the two calls and use it to find the probability that the two calls are made within 5 minutes of each other (i.e. |Y_{1} − Y_{2}| ≤ 5/60).

IX. Two random variables Y_{1}, Y_{2} have joint density

f(y_{1}, y_{2}) = \{ \begin{matrix}2e^{−y_{1}−y_{2}} & if 0 ≤ y_{1} ≤ y_{2} < ∞ \\ 0 & otherwise \end{matrix}

A) (10) Find P (Y_{2} ≤ 2 + Y_{1}). (Draw a picture!) B) (10) Find the marginal density f_{1}(y_{1}). What is the distribution of Y_{1}? C) (10) Let U = 3Y_{1} + 2Y_{2}. Find the mean μ_{U}. D) Extra Credit (10) Find the variance σ_{U}^{2} for U from part C.

X. A) (10) Let Y have a gamma distribution with α = 3.2, and β = 1. Derive a general

formula for E(Y^{k}), where k ≥ 1 is an integer. (See note 1 in the directions!) B) (10) Let Y_{1}, …, Y_{n} be independent, all with gamma distributions with the same α,

and β. Using moment-generating functions, determine the distribution of the mean \overline{Y} = \frac{1}{n} ∑_{i=1}^{n} Y_{i}.

Have a peaceful and joyous holiday season!