MATH 32 – Midterm 2 Fall Semester 2008
Duration: 50 minutes
Instructions: A cheat sheet on one side of a 8.5x11” page is allowed and must be turned in with
the exam. A calculator is allowed as well. Partial credit will be awarded for correct work, unless
otherwise specified. The total number of points is 100.
1. (25 points: 5 each) For each of the following problems, identify the type of random variable
as well as the values of all parameters. If you forget the name of the distribution, you may
write down the appropriate PDF or PMF. You do not have to write out reasoning for this
problem.
Hint: Possible answers are the following: Bernoulli(p), Binomial(n, p), Poisson(λ),
Hypergeometric(N, M, n), Uniform([a, b]), Normal(µ, σ ), Exponential(λ), Chi-Square(n), t–
distribution(n), or F–distribution(n, m).
(a) X=X1+X2where both X1and X2are independent Poisson random variables with
mean 3.
(b) His the number of heads in 40 tosses of a fair coin.
(c) A club contains 50 members; 20 are men and 30 are women. A committee of 10 mem-
bers is chosen at random. Let Wdenote the number of women on the committee.
The next two parts refer to a sample X1, X2, . . . , Xnfrom a normal population with mean 4
and variance 100.¯
X= (X1+X2+· ·· +Xn)/n is the sample mean and S2= (Pn
i=1(Xi−
¯
X)2)/(n−1) is the sample variance.
(d) The sample mean ¯
X.
(e) √n(¯
X−4)
S.
2. (20 points) Suppose that at a certain garage, the time, in minutes, required to tune up a pas-
senger car is uniformly distributed over the interval [20,45]. One day, this garage received
tune-up requests on 6 such passenger cars. What is the probability that exactly 2 of the 6 cars
take less than 30 minutes to tune up?
3. (15 points) The yearly income, X, in thousands of dollars, for families in a certain city, is
normally distributed with mean µ= 53 and variance σ2= 25. The city imposes a city
income tax of 1% for the amount above 20K. That is, the city income tax for each family
is given by the formula T= 0.01(X−20) thousands of dollars. Find the expectation and
variance of T.
4. (20 points) Find the moment generating function of the following random variable and use
your result to find the expectation and variance. The probability density function of the
random variable Xis given as
f(x) = (λe−λx, x ≥0
0,otherwise,
for some constant λ. (Xis called an exponential random variable with parameter λ.)
!!! CONTINUE ON THE BACK !!!
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