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.

3

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).

4

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.

5

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.

6

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

The length, X X , of a fish from a particular mountain lake in
Idaho is normally distributed with μ=9.6 μ = 9.6 inches and σ=1.4 σ
= 1.4 inches. (a) Is X X a discrete or continuous random
variable?
(b) Write the event ''a fish chosen has a length of less than
6.6 inches'' in terms of X X :
(c) Find the probability of this event:
(d) Find the probability that the length of a chosen fish was...

The length, X, of a fish from a particular mountain lake in
Idaho is normally distributed with μ=7 inches and σ=1.7 inches.
(a) Is X a discrete or continuous random variable? (Type:
DISCRETE or CONTINUOUS)
ANSWER:
(b) Write the event ''a fish chosen has a length equal to 4
inches'' in terms of X: .
(c) Find the probability of this event:
(d) Find the probability that the length of a chosen fish was
greater than 8.5 inches: .
(e) Find the...

(5 pts) The length, X, of a fish from a particular
mountain lake in Idaho is normally distributed with μ=8.3
inches and σ=2 inches.
(a) Is X a
discrete or continuous random variable? (Type: DISCRETE or
CONTINUOUS)
ANSWER:
(b) Write the event
''a fish chosen has a length of less than 5.3 inches'' in terms of
X: .
(c) Find the
probability of this event:
(d) Find the
probability that the length of a chosen fish was greater than 11.3...

Every day, patients arrive at the dentist’s office. If the
Poisson distribution were applied to this process:
a.) What would be an appropriate random variable? What would be
the exponential-distribution counterpart to the random
variable?
b.)If the random discrete variable is Poisson distributed with λ
= 10 patients per hour, and the corresponding exponential
distribution has x = minutes until the next arrival, identify the
mean of x and determine the following:
1. P(x less than or equal to 6)...

The exponential distribution is frequently applied
to the waiting times between successes in a Poisson
process. If the number of calls received per hour
by a telephone answering service is a Poisson random
variable with parameter λ = 6, we know that the time,
in hours, between successive calls has an exponential
distribution with parameter β =1/6. What is the probability
of waiting more than 15 minutes between any
two successive calls?

1) A company produces steel rods. The lengths of the steel rods
are normally distributed with a mean of 183.4-cm and a standard
deviation of 1.3-cm.
Find the probability that the length of a randomly selected steel
rod is between 179.9-cm and 180.3-cm.
P(179.9<x<180.3)=P(179.9<x<180.3)=
2) A manufacturer knows that their items have a normally
distributed length, with a mean of 6.3 inches, and standard
deviation of 0.6 inches.
If 9 items are chosen at random, what is the probability that...

1. The amount of gold found by miners in Alaska
per 1,000 tons of dirt follows a normal distribution with a mean of
12 ounces and a standard deviation of 2.75 ounces.
What is the probability the miners find less than 8 ounces of
gold in the next 1,000 tons of dirt excavated? Include 4 decimal
places in your answer.
2. The length of time it takes a shopper to
find a parking spot in the Costco parking lot follows...

The length of long distance phone calls, measured in minutes, is
known to have an exponential distribution with E(X)=2.
a) Find the probability that a phone call lasts less than 9
minutes
b) Find the probability that a phone call lass between 7 and 9
minutes
c) If 3 phone calls are made one after another, on average, what
would you expect the total to be?

Question 2
A statistical analysis of 1000 long-distance telephone calls
made from the headquarters of the ABC company indicates that the
length of these calls is normally distributed with population mean
is 140 seconds and standard deviation is 25 seconds.
a) What is the probability that a call lasted at least 100
seconds
b) What is the probability that a particular call lasted
between 110 and 130 seconds c) What is the length of a particular
call if only 2%...

The mean weight of loads of rock is 47.0 tons with a standard
deviation of 8.0 tons. If 24 loads are chosen at random for a
weight check, find the probability that the mean weight of those
loads is less than 46.5 tons. Assume that the variable is normally
distributed. In addition to the answer, please write out your steps
and thoughts that led you to your answer.

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