Assignment 1
Be careful to follow the Guidelines for Homework Document on Canvas/Course Documents.
1. Temple football team is playing against East Carolina on Saturday, October 6, 2018. Suppose that Temple has a 46.5% chance of winning the game. College football games cannot end in a tie.
a. What is the random variable associated with this game?
b. What is the mutually exclusive event in this case?
c. Construct a well-labeled probability distribution table based on the outcomes of this game.
2. After East Carolina, Temple will play against three other teams as listed in the table below.
|
Date
|
GAME
|
Win Probability1
|
|
October 6
|
East Carolina
|
46.5%
|
|
October 20
|
Cincinnati
|
72.8%
|
|
November 1
|
UCF
|
18.6%
|
|
November 10
|
Houston
|
65.0%
|
a. What is the probability that Temple wins ALL four games?
b. What is the probability that Temple loses ONLY ONE of the four games? Be sure to show your work. (Hint: they could lose to East Carolina AND win the others, OR they could lose to Cincinnati AND win the others ...)
1 The win probabilities are randomly assigned and refer to the probability of Temple winning the game.
3. On Thursday, ABC Grocery is expecting to receive Package A containing $150 worth of food. Based on his past experience with the delivery service, the manager estimates that this package has a chance of 10% being lost in shipment.
On Friday, ABC Grocery expects Package B to be delivered. Package B contains $400 worth of food. This package has a 30% chance of being lost in shipment.
a. What are the possible outcomes for ABC Grocery's total dollar amount of losses for package A and B? For each dollar amount of losses, describe under what circumstances it would occur. In other words, what event(s) must happen for each dollar amount of losses to occur? Please note that this asks about total dollar amount of losses, not number of losses.
b. For each of the possible outcomes you identify in part [a], derive the probability of the outcome occurring.
c. Construct [in table form] the probability distribution for total dollar amount of losses for package A and B.
d. Calculate the expected value of total dollar amount of losses. What is the unit of measurement?
e. The manager has calculated the variance for the total dollar amount of losses to be 35,625. Since you want to be sure you are using correct numbers in your evaluation, prove that the manager calculated the correct variance for total dollar amount of losses. Show all work! What is the unit of measurement?
Assignment 2
Be careful to follow the Guidelines for Homework Document on Canvas/Course Documents.
1. The Civil Air Patrol (CAP) owns airplanes used for search and rescue missions. During their activities, CAP sustains losses in the form of property damage to their airplanes.
The CAP owns 150 planes in Pennsylvania (PA), and it has the following data for the number of accidents per airplane per year in PA.
|
# of accidents per airplane per year
|
# of airplanes of having this # of accidents
|
|
0
|
85
|
|
1
|
45
|
|
2
|
20
|
a. What is the random variable illustrated here?
b. Derive the probability distribution based on this data. Be careful to label exactly what you are defining as the probability distribution. Round your calculation to 2 decimals.
c. Calculate the expected value of frequency per airplane. What is the unit of measurement? Round your calculation to 2 decimals.
d. CAP has calculated the variance of frequency per airplane to be equal to 0.5064 in PA. Prove that CAP calculated the correct variance for frequency. Show all work! What is the unit of measurement?
e. How much risk is present for CAP in PA? What is the unit of measurement? Round your calculation to 2 decimals.
2. Assume initially that when accidents occur, they are non-random and equal to $13,000.
a. Calculate the expected loss per airplane in PA.
b. Calculate the expected loss for ALL airplanes in PA.
3. Now based on the information in PA, CAP has constructed the following information related to the dollar amount of losses for 105 reported accidents.
a. Calculate the expected value of severity per airplane. Round your calculation to 2 decimals.
|
$ amount of losses
|
# of accidents having
this $ amount of losses
|
|
$500
|
7
|
|
$1,000
|
10
|
|
$3,000
|
2
|
|
$6,000
|
6
|
|
$9,000
|
8
|
|
$13,000
|
18
|
|
$16,000
|
13
|
|
$19,000
|
15
|
|
$23,000
|
9
|
|
$26,000
|
6
|
|
$29,000
|
4
|
|
$33,000
|
3
|
|
$36,000
|
1
|
|
$39,000
|
2
|
|
$50,000
|
1
|
|
$80,000
|
0
|
b. Calculate the expected loss per airplane.
4. The CAP also has the following data for the number of accidents per airplane in New Jersey (NJ). CAP has 60 airplanes in NJ. It is known that the variance of frequency per airplane in NJ is 0.2989.
|
# of accidents per
airplane per year
|
# of airplanes of having
this # of accidents
|
|
0
|
40
|
|
1
|
18
|
|
2
|
2
|
Which state is riskier for CAP, PA or NJ? And why? Show all calculations and explain your numerical results. Round your calculation to 2 decimals.