MATH 32 – Final Exam Spring Semester 2009
Duration: 3 hours
Instructions: Answer all questions, without the use of notes or books. Calculators may be used to
calculate numbers only. Partial credit will be awarded for correct work, unless otherwise speci-
fied. The total number of points is 100.
1. (7 points) Suppose that X1, . . . , Xnare independent exponential random variables with pa-
rameter λ. That is, they all have the same probability density function
f(x) = (λe−λx, x ≥0
0,otherwise.
Determine the maximum likelihood estimator of λ. Show your argument.
2. (8 points total) You asked your neighbor to water a sickly plant while you are on vacation.
Without water it will die with probability .8; with water it will die with probability .15. You
are 90% certain that your neighbor will remember to water the plant.
(a) (5 points) What is the probability that the plant will be alive when you return?
(b) (3 points) If it is dead, what is the probability your neighbor forgot to water it?
3. (8 points) A sample of 20 cigarettes is tested to determine nicotine content and the average
value observed was 1.2 mg.
(a) (4 points) Compute a 99 percent two–sided confidence interval for the mean nicotine
content of a cigarette if it is known that the standard deviation of a cigarette’s nicotine
content is σ=.2mg.
(b) (4 points) Suppose that the population variance is not known in advance of the experi-
ment. If the sample variance is .04, compute a value cfro which we can assert ”with 99
percent confidence” that cis lager than the mean nicotine content of a cigarette.
4. (7 points) The lifetime of a certain electrical part is a random variable with mean 100 hours
and standard deviation 20 hours. If 16 such parts are tested, use the Central Limit Theorem
to approximate the probability that the sample mean is less than 104.
5. (12 points: 3 each) A college professor never finishes his lecture before the end of the hour
and always finishes his lectures within 2 min after the hour. Let X=the time, measured in
minutes, that elapses between the end of the hour and the end of the lecture and suppose
the probability density function of Xis
f(x) = (kx20≤x≤2
0otherwise .
(a) Find the value of k.
(b) What is the probability that the lecture continues beyond the hour for between 1 and
1.5 min?
(c) Calculate E[X].
(d) Calculate Var(X).
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