Mathematics 375 – Probability and Statistics I Makeup Exam 2 – December 11, 2009

Directions:

Do all work in the blue exam booklet. There are 100 possible points and 10 Extra Credit points (Please turn over for question IV and the Extra Credit problem.)

I. Birth weights of babies born in the US are normally distributed with mean μ = 3315 grams and standard deviation σ = 575 grams.

A) (5) What is the probability that a baby will have birth weight more than 3890 grams? B) (7.5) What is the probability that a baby will have birth weight between 2383.5 grams

and 4005 grams? C) (7.5) If 30 newborn babies are selected randomly, what is the probability that 6 of

them weigh between 2383.5 grams and 4005 grams? D) (5) If we knew the mean and standard deviation, but we did not know the distribution

of the birth weights, what could we say in general about the probability in part A?

II. A) (10) Twelve out of a batch of twenty identical-looking packets of white crystals contain

sugar and five contain artificial sweetener. Three packets are randomly selected, found to contain sugar, and discarded. What is the probability that if four packets are selected from those remaining, no more than one contains sugar?

B) (10) The number of customers arriving at a sandwich shop to place orders has a Poisson distribution with mean 14 per hour. Give a formula for computing the probability that at least 20 customers arrive in a given hour. (It is not necessary to obtain a decimal approximation for this number.)

III. A) (5) Let

g(x) = \{ \begin{matrix}\frac{1}{3}e^{x/3} & if x < 0 \\ 0 & otherwise. \end{matrix}

Show that g is a legal pdf. (This is a “reflected exponential.”)

For the remainder of this problem let

f(y) = \{ \begin{matrix}\frac{1}{6}e^{y/3} & if y < 0 \\ \frac{1}{2}ye^{−y} & if y > 0 \end{matrix}

B) (10) Show that f(y) is a legal pdf for a random variable Y. C) (10) Find E(Y). (Don’t work harder than necessary!) D) (10) Determine the moment-generating function of Y. On which interval of t values

is your formula valid?

IV. Let Y_{1}, Y_{2} be jointly continuous random variables with joint density

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

A) (10) What is the value of c? B) (10) What is P (Y_{1} ≤ 3/4, Y_{2} ≤ 1/2)?

Extra Credit. (10) If Y has a standard normal distribution, then X = e^{Y} is said to have a lognormal distribution. Determine E(X) and V (X) for X with a lognormal distribution.