Mathematics 375 – Probability and Statistics I Midterm Exam 2 – November 19, 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 less than 2165 grams? B) (5) What is the probability that a baby will have birth weight between 2866.5 grams

and 3763.5 grams? C) (10) If newborn babies are selected randomly, how many babies would you expect to

have to weigh to find the first one that weighs between 3763.5 grams and 4258 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) Fifteen out of a batch of twenty identical-looking packets of white crystals contain

sugar and five contain artificial sweetener. Four packets are randomly selected, found to contain sugar, and discarded. What is the probability that if three packets are selected from those remaining, at least one contains sugar?

B) (10) The number of customers arriving at a coffee shop to place orders has a Poisson distribution with mean 20 per hour. Give a formula for computing the probability that more than 40 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}{2}e^{x/2} & 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}{4}e^{y/2} & if y < 0 \\ \frac{1}{18}ye^{−y/3} & if y > 0 \end{matrix}

B) (10) Show that f(y) is a legal pdf for a random variable Y. C) (10) Show that E(Y) = 2. (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_{2} & if 0 ≤ y_{1} ≤ 1 and y_{1}^{2} ≤ y_{2} ≤ 1 \\ 0 & otherwise \end{matrix}

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

Extra Credit. (10) Given two real numbers a, b with a < 0 < b, it is possible to find a random variable U whose moment-generating function is

m_{U}(t) = \frac{1}{(1 − at)(1 − bt)}

for some interval of t in R? If so, give a formula for the pdf for such a random variable. If not, say why not. (Hint: Think about what happened in problem III above.)