Mathematics 375 – Probability Theory Midterm Exam 2 – November 17, 2011

Directions: Do all work in the blue exam booklet. There are 100 possible points and 10 Extra Credit points.

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) Find the probability that a randomly chosen baby will have birth weight more

than 3700 grams. B) (10) If 20 newborn babies are selected randomly, how many of them would be expected

to have weights between 2740 and 3890 grams? C) (10) If we knew the mean and standard deviation, but we did not know the precise

distribution of the birth weights, what could we say in general about the probability that a randomly selected baby has weight differing from 3315 grams by at least 862.5 grams?

II. (15) The number of defects Y in a 1-foot segment of a magnetic tape has a Poisson distribution with mean 3. What is the probability that a 1-foot segment contains more than 4 defects?

III. (10) (From an actuarial “P Exam” problem) The warranty on a machine specifies that it will be replaced at failure or at age 4, whichever occurs first. The machine’s age at failure, X, is a random variable with probability density function

f(x) = \{ \begin{matrix}1/5 & if 0 < x < 5 \\ 0 & otherwise \end{matrix}

Let Y be the age of the machine at replacement. What is E(Y)?

IV. Let

g(x) = \{ \begin{matrix}\frac{1}{16}x^{2}e^{x/2} & if x < 0 \\ 0 & otherwise (that is, if x ≥ 0). \end{matrix}

A) (10) Show that g is a legal pdf for a random variable X. B) (10) Determine the moment-generating function of X. On which interval of t values

is your formula valid?

V. 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 y_{1}^{2} ≤ y_{2} ≤ 1 \\ 0 & otherwise \end{matrix}

A) (15) What is P (Y_{1} ≤ Y_{2})? (Note that you will need to determine c.) B) (15) Find the marginal density f_{1}(y_{1}) and use it to compute V (Y_{1}).

Extra Credit. (10) What is E(X) for the random variable X from question IV?