Mathematics 375 – Probability and Statistics I Final Examination – December 14, 2009

Directions

Do all work in the blue exam booklet. There are 200 possible regular points and 10 possible Extra Credit points. If an answer can be expressed in terms of binomial or multinomial coefficents, factorials, the gamma function, etc. please leave it in that form rather than evaluating. Even if you cannot see how to completely solve a problem, do not leave it blank. Partial credit will be given for relevant definitions, ideas that could lead to solutions, and so forth.

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

3, 3, 2, 4, 1, 0, 1, 2, 3, 5, 1, 2, 2, 1, 1, 0, 2, 4, 3, 1

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

integer value. B) (10) How many of the data values are within two sample standard deviations of the

sample mean? How does this compare with the “empirical rule?”

II. Suppose you have a key ring with N keys, exactly one of which is your car key. You are parked in a country lane on a moonless night and can’t see the keys or the ring. So you try the keys in your car door lock one by one until you find the right one.

A) (10) Assuming you are not careful and you mix up all the keys after each try, what is

the probability that you take at least y tries to find the right key? B) (5) Let Y be the discrete random variable giving the number of the trial on which you

find the right key. What is the expected value of Y? C) (10) Now suppose you are more careful and you take each wrong key off the key ring

and put it aside after you try it so that it does not get mixed back in with the keys you have not tried. Let Z be the discrete random variable giving the number of the trial on which you find the right key by this method. What is the expected value of Z?

III. (Hypothetical) In the city of Worcester, 50 percent of registered voters are Democrats, 30 percent are Republicans, and 20 percent are Independents. In the recent Senate primary, 15 percent of Democrats voted, 10 percent of Republicans voted, and 35 percent of Independents voted.

A) (10) If a single voter is selected at random, what is the probability that he or she

voted in the Senate primary? B) (10) Given that a registered voter did vote, what is the probability that the voter was

an Independent?

IV. Let X be a random variable for which E(X) = μ and V (X) = σ^{2}. Let c be an arbitrary constant. A) (10) Show that E((X − c)^{2}) = (μ − c)^{2} + σ^{2}. B) (5) For which c is E((X − c)^{2}) a minimum?

V. The number of customers arriving at a fast food restaurant in a typical hour has a Poisson distribution with mean 20. A) (5) What is the probability that 17 customers arrive in a given hour? B) (10) If the store is open 12 hours a day, what is the probability that exactly 17

customers will arrive in exactly 4 out of the 12 hours?

VI. A continuous random variable Y has p.d.f

f(y) = \{ \begin{matrix}c(y^{2} + y) & if 1 ≤ y ≤ 2 \\ 0 & otherwise \end{matrix}

A) (5) What is the value of c? B) (5) What is P (1 ≤ y ≤ 3/2)? C) (10) Find E(Y) and V (Y).

VII. (15) The diameters of the bolts in a large box follow a normal distribution with mean 2 cm and a standard deviation of 0.03 cm. The diameters of the holes in the nuts in another large box are also normally distributed, but with mean 2.02 cm and standard deviation 0.04 cm. A bolt and a nut will fit together if the hole in the nut is larger than the diameter of the bolt, but the difference is no larger than .05 cm. If a nut and a bolt are selected independently and randomly from the two boxes, what is the probability that they will fit together?

VIII. Let X and Y be independent random variables with moment-generating functions

m_{X}(t) = m_{Y} (t) = e^{t^{2}+3t}

A) (10) What is the moment generating function of Z = 3X + 2Y − 4? B) (10) What is the distribution of Z?

IX. Twenty students in a probability and statistics class take a final examination independently of one another. The number of minutes each student requires to complete the exam is a random variable with an exponential distribution with mean 120 minutes. The students all start work at 8:30am. A) (10) What is the probability that at least one student out of the 20 will complete the

exam by 10:00 am? B) (10) What is the distribution of the total time taken by all of the students to complete

the exam?

X. Suppose Y_{1}, Y_{2} have joint density

f(y_{1}, y_{2}) = \{ \begin{matrix}24y_{1}y_{2} & if y_{1} ≥ 0, y_{1} + y_{2} ≤ 1 \\ 0 & otherwise \end{matrix}

A) (10) Are Y_{1}, Y_{2} independent? B) (10) What is V (8Y_{1} − 2Y_{2})? C) (10) Use the method of distribution functions to find the density function for U =

Y_{1} + Y_{2}.

Extra Credit (10) One form of the Law of Large Numbers states that if X_{1}, …, X_{n} are independent and identically distributed random variables for which E(X_{i}) = μ and V (X_{i}) = σ^{2} exist, then for any ϵ > 0,

lim_{n→∞} P (|\overline{X} − μ| < ϵ) = 1.

Prove this statement without assuming anything else about the distribution of the X_{i}.