Question

Solutions for this exercise will not be posted. However, it is possible that questions from this
exercise could appear on Midterm II. DEFINE ALL NOTATION!!!!!

1. Here is a pdf:
.
a) How do you know it is a continuous distribution?

b) The constant a is positive. What is a?

c) What is probability that the random variable X is equal to 1?

d) What is F(-0.5)?

e) What is the cdf of the random variable X?

f) What is E(X)?

g) What is V(X)?

h) Find the median of X.

i) Find the 20th percentile of X

2

2. The amount sold of a pharmaceutical product per month has the uniform distribution between
9 and 11 pounds. The profit obtained is 8 times the amount sold–squared. For example, if
9.7 pounds are sold, then the profit is 8*9.7 2 . What is the mean and variance of the profit?

3. The dimension of a part made by a manufacturer is normally distributed with mean 20 and
standard deviation 1.2.
a) Find the 2 th percentile of the part dimension.

b) A customer has specifications 20.5 +/- 1.5. What will be the fraction out-of-specification
if the customer buys these parts?

c) The manufacturer can reset the mean of the process. What should the mean be set to such
that 98 percent of parts have dimension within the customer specifications. Assume the
sd remains at 1.2.

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4. The number of calls that arrive to a call center is Poisson distributed with mean 4.1 per
minute.

a) What is the probability of less than 2 calls per minute.

b) What is the mean time between calls?

c) Write the pdf of the time between calls?

d) What is the mean time between calls in seconds?

e) What is the probability that the time to the next call is less than a second?

f) No calls have come in to the call center for the last 5 minutes – 300 seconds. Use
conditional probability to demonstrate the memoryless property and show that the
probability the next call arrives within a second matches your answer to part e).

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5. Create a problem where the given is about a random variable that is exponential. Ask a
question that requires the exponential distribution & solve. Ask a question that requires the
use of the Poisson & solve. (Note – problem 4 gives information about a Poisson random
variable and then asks Poisson and exponential questions.)

6. One piece of PVC pipe is to be inserted inside another piece. The overlap should be
normally distributed with mean 1 inch and sd 0.1. The lengths of the pieces are independent
and normally distributed: for piece 1 the mean is 20 and sd 0.5 inches; for piece 2 the mean is
15 and sd 0.4 inches. Find the probability that the total length after insertion is between 34.5
and 35 inches.

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7. Each section of fencing is normally distributed with mean length equal to 6 feet and
standard deviation 0.3 feet.
a) If 10 sections of fencing are installed, end to end, find the mean and standard deviation
of the total length in feet.

b) What is the mean and standard deviation of the total length in inches?

8. Create a clearance problem and solve it.

9. Create a profit problem and solve it.

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10. The weight on a bridge is estimated to have the uniform distribution between 3 and 5 tons.
At 16 random times I record the vehicles on the bridge, and figure out the weight. I then
compute the average of the 16 observations.

a) Define a random variable that is W = the weight on the bridge at observation i, i=1…16.
Give the distribution, specify the parameters and give the expected value and variance.

b) Define another random variable =sample average of 16 observations of weight on the
bridge. Give the distribution, specify the parameters and give the expected value and
variance.

c) Find the probability that the sample average exceeds 4.5 tons.

d) Find the probability that the weight on the bridge at any given time exceeds 4.5 tons.

e) In each of the parts of question 10, show where you invoked the Central Limit Thm.

Answer 1

dear student, please post the questions one at a time.

The PDF for the 1. is missing hence I am attempting question 2.

2) For uniform distribution between A & B

the mean:

Variance

The amount sold of a pharmaceutical product per month has the uniform distribution between
9 and 11 pounds.

Hence the mean amount sold of a pharmaceutical product per month =

the variance of the amount sold of a pharmaceutical product per month :

The profit obtained is 8 times the amount sold–squared.

Hence the Mean of the profit is = 8 * the mean amount sold of a pharmaceutical product per month = 8*10 = $80

The variance of the profit is = the variance of the amount sold of a pharmaceutical product per month = $21.33