Mathematics 375 – Probability Theory Final Examination – December 16, 2011

Directions

You have as much time as you need to work on this exam. Do problems I - VIII 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 \\ 0.44 & 0.48 & 0.53 & 0.54 & 0.55 & 0.66 & 0.69 & 0.70 & 0.72 & 0.73 \\ 0.73 & 0.74 & 0.74 & 0.76 & 0.77 & 0.78 & 0.81 & 0.95 & 0.97 & 1.02 \\ 1.03 & 1.06 & 1.07 & 1.13 & 1.13 & 1.14 & 1.19 & 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 a relative frequency histogram for this data using 10 equal bins on the interval [0, \frac{3}{2}].

B) (20) Two models are proposed for the distribution of Y:

1. a uniform distribution on [0, \frac{3}{2}], or

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

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

Find the expected values and variances of the two proposed model distributions.

C) (5) The actual sample mean is \overline{y} = .8542 and the sample standard deviation is s ≐ .4033. Is either of the models in part B clearly more likely as the underlying distribution from which the values in part A were generated? Explain.

II. (15) A bin contains five components from supplier A, three from supplier B, and four from supplier C. If three of the components are selected randomly (without replacement) for testing, what is the probability that the components chosen will come from at most two of the suppliers?

III. Rutabaga seeds from grower A have an 90% germination rate, while seeds from grower B have a 80% germination rate and those from grower C have a 85% germination rate. A seed packaging company buys 25% of its seeds from grower A, 35% from grower B, and 40% 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 discrete random variable representing 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 no element of A to itself?

V. A candy maker produces candy bars that have a stated label weight of 57.4 grams. The manufacturing process is subject to some randomness, though, and the actual weights of the bars are normally distributed with mean μ = 58 grams and σ = 2 grams.

A) (5) What is the probability that a single candy bar has weight < 57 grams?

B) (5) If candy bars are selected independently and at random from the production line, what is the probability that at least 4 trials will be necessary to find one that has weight < 57 grams?

C) (5) If 20 candy bars are selected independently and at random from the production line, what is the probability that 3 or fewer will have weight < 57 grams?

D) (10) The production process can be changed to alter the average weight μ of the bars, but σ = 2 cannot be changed. What value for μ will make the probability that a candy bar weighs < 57.4 grams equal to .01?

VI.

A) (10) State the Central Limit Theorem.

B) (10) From experience, a charity knows that the contributions it receives have mean μ = 3125 and σ = 250. If n = 2000 of the contributions are selected at random, estimate the probability that the average of those 2000 contributions is > 3200.

VII. Let Y be a random variable with density function

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

A) (10) Determine the value of k, and find E(Y) and V (Y).

B) (10) Let Y_{1}, …, Y_{n} be independent random variables all with density as above. What is the distribution of \overline{Y} = \frac{1}{n}(Y_{1} + ⋯ + Y_{n})?

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

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

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}).

D) (10) Let U = Y_{1}^{2}. Determine the pdf for U.

Extra Credit. (20) A volcano is located at the center (x, y) = (0, 0) of a circular island 1 mile in radius. When the volcano erupts, it emits lava that spreads over a random sector of the island. The diagram below shows one such sector between two radii drawn with dashes. The middle radius of the sector (solid line in diagram) makes a random angle with the positive x-axis, Θ, which is uniformly distributed on the interval [−π, π]. The total central angle of the sector (the angle between the two dashed radii in the diagram) is a second random variable Φ that has pdf

f(φ) = \begin{cases}c(π/2 − φ) & if 0 ≤ φ ≤ π/2 \\ 0 & otherwise \end{cases}

for some constant c. The angles Θ and Φ are independent.

You own a villa at (x, y) = (1, 0) on the coast of the island. What is the probability that any single eruption buries your villa in lava?

Have a peaceful and joyous holiday season!