Mathematics 375 – Probability and Statistics 1 Final Examination – December 18, 2003

Directions

You have until 5:30 p.m. to work on this exam. Do problems I - IX 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 binomial coefficients, the gamma function, factorials, etc., it is preferable to leave it in that form.

2) It is OK (even suggested!) to use general formulas we have derived, etc. rather than working out probability integrals, etc. from scratch.

3) 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. A large number of observations of a certain continuous random variable Y were taken, and a typical subset of n = 50 of them were ordered to produce the data set below:

\begin{matrix}0.10 & 0.23 & 0.25 & 0.26 & 0.26 & 0.28 & 0.31 & 0.39 & 0.42 & 0.44 \\ 0.45 & 0.48 & 0.53 & 0.54 & 0.55 & 0.66 & 0.69 & 0.70 & 0.72 & 0.73 \\ 0.73 & 0.74 & 0.74 & 0.75 & 0.77 & 0.78 & 0.81 & 0.95 & 0.97 & 1.02 \\ 1.03 & 1.05 & 1.07 & 1.13 & 1.13 & 1.14 & 1.20 & 1.22 & 1.27 & 1.28 \\ 1.30 & 1.34 & 1.34 & 1.37 & 1.39 & 1.40 & 1.43 & 1.43 & 1.46 & 1.48 \end{matrix}

A) (10) Construct relative frequency histograms for this data using 5 bins on the interval

[0, 1.5], and then 10 bins on the interval [0, 1.5]. B) (10) Two models are proposed for the distribution of Y:

1. a uniform distribution on [0, 1.5], or

2. a distribution described by the probability density function (pdf)

f(y) = \{ \begin{matrix}\frac{8y}{9} & if 0 ≤ y ≤ 3/2 \\ 0 & otherwise \end{matrix}

Find the expected values and variances of the two proposed model distributions. C) (10) The actual sample mean is \overline{y} = .8542 and the sample standard deviation is

s ≐ .4033. How well does the “empirical rule” match the distribution of Y?

II. (20) A bin contains three components from supplier A, four from supplier B, and five from supplier C. If four of the components are selected randomly for testing, what is the probability that each supplier will have at least one component tested?

III. Carrot seeds from grower A have an 85% germination rate, while seeds from grower B have a 75% germination and those from grower C have a 90% germination rate. rate. A seed packaging company buys 30% of its seeds from grower A, 40% from grower B, and 30% from grower C, then mixes them thoroughly in making up packets for sale.

A) (10) What is the probability that a randomly selected seed will germinate? B) (10) Given that a seed did not germinate, what is the probability that it came from

grower B?

IV. A permutation of A = {1, 2, 3, 4, 5} is a one-to-one, onto mapping from this set to itself. For instance, the f defined by

f(1) = 2, f(2) = 4, f(3) = 3, f(4) = 1, f(5) = 5

is one such permutation. Consider a permutation selected at random, and let Y be the number of elements of the set A that are mapped to themselves. For instance, if the f above was selected, then the value of Y is 2 (since f(3) = 3 and f(5) = 5). For this problem, you are given the information that the moment-generating function of Y is

m(t) = \frac{44}{120} + \frac{45}{120}e^{t} + \frac{20}{120}e^{2t} + \frac{10}{120}e^{3t} + \frac{1}{120}e^{5t}

A) (10) Find the mean and variance of Y. B) (10) What is the probability that a randomly selected permutation maps at least one

element of A to itself?

V. A candy maker produces thin chocolate mints (yum!) that have a label weight of 20.4 grams. The manufacturing process is subject to some randomness, though, and the actual weights of the mints are normally distributed with mean μ = 20.8 and σ = 0.3. A) (10) What is the probability that a single mint has weight < 20.4 grams? B) (10) If mints are selected independently and at random from the production line, what

is the probability that at least 10 trials will be necessary to find one that has weight < 20.4 grams?

C) (10) If 40 mints are selected independently and at random from the production line, what is the probability that 3 or fewer will have weight < 20.4 grams?

VI. (10) 900 students are given an exam. The average score is \overline{y} = 83 and the sample variance is v = 36. At least how many students must have scored between 71 and 95?

VII. Telephone calls arrive at a switchboard.

A) (10) The number of calls within a 10 minute interval has a Poisson distribution with mean λ = 5. What is the probability that more than 10 calls arrive in a 10 minute interval?

Starting from any instant, the elapsed time Y until the 5th call arrives is a random variable with a pdf of the form

f(y) = \{ \begin{matrix}ky^{4}e^{−2y} & if y > 0 \\ 0 & otherwise \end{matrix}

B) (10) Determine the value of k. C) (10) Find E(Y) and V (Y).

D) (10) The elapsed time Y until the 5th call and the elapsed time Z between the 5th and 10th calls are independent, and both have the same distribution. Find the distribution of the random variable Y + Z by using the appropriate moment-generating functions.

VIII. Let Y_{1} and Y_{2} be random variables with joint pdf

f(y_{1}, y_{2}) = \{ \begin{matrix}3y_{1}^{2}y_{2} & if −1 ≤ y_{1} ≤ 1, 0 ≤ y_{2} ≤ 1 \\ 0 & otherwise \end{matrix}

A) (10) What is P (Y_{2} > Y_{1})? (Hint: Draw a picture!) B) (10) Are Y_{1} and Y_{2} independent? Why or why not? C) (10) Determine V (3Y_{1} + 2Y_{2}).

Extra Credit (20)

Let Y be a continuous random variable whose cumulative distribution function F (y) satisfies F (y) = 0 for all y ≤ 0. Assume that Y has the “memoryless” property:

P (Y > a + b|Y > a) = P (Y > b)

for all a, b > 0. Show that g(y) = 1 − F (y) satisfies

g(y + y′) = g(y)g(y′)

for all y, y′ > 0. What type of distribution does Y have?

Have a peaceful and joyous holiday season!