Mathematics 375 – Probability and Statistics I Midterm Exam 2 – November 19, 2003

Directions: Do all work in the blue exam booklet. There are 100 possible points. You may leave answers in terms of of factorials, binomial coefficients, etc. if appropriate.

I. (15) Ten college students, exactly four of whom are not of legal age, go out to a bar and all order alcoholic beverages. The waitperson taking their order selects 5 students at random from the group to “card” and must refuse to serve any one who is under legal age. What is the probability that exactly 2 students will be refused service? (Assume, unrealistically perhaps, that each ID shows that student’s actual age!)

II. The fraction Y of impurities in a tank-car lot of industrial grade potassium perchlorate is a random variable with probability density function

f(y) = \{ \begin{matrix}cy^{2}(1 − y)^{2} & if 0 < y < 1 \\ 0 & otherwise \end{matrix}

A) (10) What is the value of the constant c? B) (10) What is the variance of Y? C) (10) Find the cumulative distribution function for Y. D) (10) Find P (.1 ≤ Y ≤ .25).

III. The life in years of a certain type of electrical switch has an exponential distribution with mean 2 years. A) (15) What is the probability that a single switch will fail during the first year after it

is installed? B) (5) If 100 of these switches are installed in different, independently operating systems,

what is the probability that at most 30 fail during the first year? (You need not obtain a single decimal approximation to this number; a formula for computing it will suffice.)

IV. The weights of a population of miniature poodles are normally distributed with mean μ = 8kg and standard deviation σ = .9kg. A) (15) What is the probability that a randomly selected poodle will have weight between

7.3kg and 9.1kg? B) (10) Let Y be the weight of a poodle from the population as a random variable.

What is E(Y^{3})? (Note: part credit will be given for a correct verbal description of a procedure that could be used to derive this value even if you cannot carry out the procedure completely.)

Extra Credit (10) Let Y be a random variable with probability density function

f(y) = \{ \begin{matrix}ye^{−y} & if y > 0 \\ 0 & otherwise \end{matrix}

Find the moment-generating function m_{Y} (t), assuming t < 1.