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1. I recently had to replace both front headlights on my car. The life expectancy of my
headlights follows an exponential distribution with a MTBF of 1500 hours. That is, the
expected number of hours until failure is 1500 hours. For the purposes of this problem
assume that both headlights were initially installed at the same time, are identical models
and are always on at the same time.
a. What is the probability that the right headlight will burn out in 950 hours or less of
operation?
b. What is the probability that both headlights burn out in 950 hours or less of operation?
th
c. Sure enough the right headlight burns out at the 950 hour of operation. Now suppose
that I get the right headlight replaced. As I am driving some time later, I nervously
await the right headlight failing after 950 hours but nothing happens. I start feeling
lucky and believe that the right headlight will last at least another 950 hours. What is
the probability that my right headlight will last for at least 1900 hours given that it
has not failed after 950 hours?
2. Use Excel to illustrate the Central Limit Theorem when sampling from a binomial
distribution.
a. Generate at least 1000 random samples (you can generate more if you wish), each with 4
observations, from a binomial random variable y where p=0.2 (the probability of a
success), and n=10 trials. That is, the random variable y represents the number of
successes in 10 trials, where the probability of a success is 0.2. Calculate the sample
mean for each sample.
• Plot a frequency histogram of the sample means and comment on the plot.
• Determine the range, mean and standard deviation of the 1000 sample means.
• How do the calculated mean and standard deviation compare with what you
would have expected from a theoretical analysis?
b. Generate at least 1000 random samples (you can generate more if you wish), each with 30
observations, from a binomial random variable y where p=0.2 (the probability of a
success), and n=10 trials. That is, the random variable y represents the number of
successes in 10 trials, where the probability of a success is 0.2. Calculate the sample
mean for each sample.
• Plot a frequency histogram of the sample means and comment on the plot.
• Determine the range, mean and standard deviation of the 1000 sample means.
• How do the calculated mean and standard deviation compare with what you
would have expected from a theoretical analysis?
3. The CARS.XLS file that came with your Berk & Carey textbook contains information on
392 car models. Data in the file includes the miles per gallon (MPG) of each car, as well
as the acceleration, weight and horsepower.
a. Create two scatterplots, the first between MPG and acceleration and the second between
weight and horsepower. Describe the relationships you see.
b. Find the standard deviation and coefficient of variation for MPG, acceleration, weight and
horsepower, respectively. Interpret the results, paying particular attention to the units of each
measure.
c. Find the correlation and covariance between a car’s MPG and its acceleration. Interpret the
results, paying particular attention to the units of each measure.
d. Find the correlation and covariance between a car’s MPG and its weight. Interpret the
results, paying particular attention to the units of each measure.
e. Find the correlation and covariance between a car’s MPG and its horsepower. Interpret the
results, paying particular attention to the units of each measure.
4. A certain market has both an express checkout line and a super-express checkout line. Let
Y denote the number of customers in line at the express checkout at a particular time of
day, and let X denote the number of customers in line at the super-express checkout at the
same time. Suppose the joint probability mass function of Y and X is as given in the
accompanying table.
X
p(x,y)
Y
0
1
2
3
4
0
0.08
0.06
0.05
0
0
1
0.07
0.15
0.04
0.03
0.01
2
0.04
0.05
0.1
0.04
0.05
3
0
0.04
0.06
0.07
0.06
a. What is the probability that the number of customers in line at the express checkout is 3?
b. Are X and Y independent random variables?
c. What is the probability that there is exactly one customer in each line at the same time?
d. What is the probability that the number of customers in the two lines are identical?
e. You are interested in examining whether the express checkout handles more customers
than the super-express checkout. Generate the probability mass function of the random
variable A, where A = Y – X.
f. What is the probability that the express checkout has more customers than the superexpress checkout, that is, the p(A > 0).
5. An airport limousine can accommodate up to four passengers on any one trip. The
company will accept a maximum of six reservations for a trip and a passenger must have
a reservation. From previous records, 20% of all those making reservations do not appear
for the trip. Answer the following questions, assuming independence whenever
appropriate.
a. If six reservations are made, what is the probability that at least one individual with a
reservation cannot be accommodated on the trip?
b. Suppose that the probability mass function of the number of reservations made is given in
the accompanying table. Let X denote the number of passengers on a randomly selected
trip. Obtain the probability mass function of X.
Number of
reservations
Probability
3
4
5
6
0.1
0.2
0.3
0.4
6. (15 Points) Consider the clearance, C, between a shaft and a bearing. The shafts are
mass produced such that the outside diameters, S, are normally and independently
distributed with mean μs = 1.05 inches, and a standard deviation σs. The bearings are
produced such that the inside diameters, B, are normally and independently
distributed with mean μ = 1.06 inches and a standard deviation σ = 0.001 inch. If
b
b
production of shafts is to be controlled so that no more than 1% of randomly mated
shafts and bearings will exhibit interference, what is the maximum allowable value of
σs?
Note that C = B – S.