Mathematics 375 – Probability and Statistics I Midterm Exam 1 – October 8, 2009

Directions:

Do all work in the blue exam booklet. There are 100 possible points and 10 Extra Credit points (please turn over for the Extra Credit problem).

I. To test the effectiveness of a seal for air leaks in automobile tires, after the seal was installed in a tire, a needle was inserted into the tire and air pressure was increased until leakage was observed. The pressures (in lb per square inch) where leakage first occurred on 10 trials were:

93, 87, 91, 90, 92, 88, 95, 91, 90, 96

A) (10) Construct a relative frequency histogram for this data on the interval [86.5, 96.5], subdividing into 5 equal “bins”

(86.5, 88.5], (88.5, 90.5], (90.5, 92.5], (92.5, 94.5], (94.5, 96.5].

B) (15) How many of the data points are within 1 standard deviation of the sample mean?

II. Sailors on a ship send signals by arranging (exactly) 8 colored flags along a rope line. A) (5) How many different signals can they send if they have flags of 8 different colors? B) (10) How many different signals can they send if they have 4 red, 2 green, and 2 blue

flags? (There are no differences between the flags of the same color.) C) (10) In the situation of part B, if a random arrangement of flags is constructed, what

is the probability that at least 5 other flags appear between the two blue flags?

III. Let A_{1}, A_{2}, A_{3} be events in a sample space S. Assume that S = A_{1} ∪ A_{2} ∪ A_{3}, where A_{i} ∩ A_{j} = ∅ if i ≠ j. Let P (A_{1}) = .35, P (A_{2}) = .4, P (A_{3}) = .25. Finally, let B be another event with P (B|A_{1}) = .1, P (B|A_{2}) = .2 and P (B|A_{3}) = .05. A) (10) State the Law of Total Probability and use it to compute P (B). B) (5) Are B and A_{2} independent events? Why or why not? C) (10) What is P (A_{1} ∪ A_{2}|B)? Give the name(s) of any general rule(s) that you are

using.

IV. A) (10) Let Y have a binomial distribution with parameters n = number of trials and

p = success probability on each trial. Show that E(Y) = np. B) (15) An army regiment has 20 different squads: 16 with 25 soldiers each, 3 with 100

soldiers each, and one with 300 soldiers, for a total of 1000 soldiers. Select a soldier at random out of the 1000 and let the random variable X equal the size of the squad to which that soldier belongs. Find the probability mass function, the expected value, and the variance of X.

Extra Credit (10)

Let X be a discrete random variable which takes as values all natural numbers k ≥ 1. Assuming that E(X) exists, show that

E(X) = ∑_{k=1}^{∞} P (X ≥ k).

(Also, why do we need to add the hypothesis that E(X) exists in a case like this?)