1. The Amazon, Inc. owns trucks used for cargo shipment. In the course of their activities, Amazon sustains losses in the form of property damages to their trucks. Assume that the Amazon owns a truck station located in Pennsylvania (PA) and also another station in New Jersey (NJ). The PA station has 300 trucks. The Amazon has the following data for the number of accidents per truck in the PA station.
|
# of accidents per truck per year
|
# of trucks of having this # of accidents
|
|
0
|
150
|
|
1
|
87
|
|
2
|
63
|
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. [2 points]
c. Calculate the expected value of frequency per truck. What are the units of measurement?
d. Amazon has calculated the variance of frequency per truck to be equal to 0.6259 for the PA station. Prove that Amazon calculated the correct variance for frequency. Show all work! What are the units of measurement?
e. How much risk is present for the PA station? What are the units of measurement?
2. Assume initially that when accidents do occur, they are non-random and equal to $2000.
a. Calculate the expected loss per truck for PA truck station.
b. Calculate the expected loss for ALL trucks in PA truck station.
3. Now based on the information in PA station, Amazon has constructed the following information related to the dollar amount of losses for 213 reported accidents.
|
S amount of
losses
$300
|
# of accidents having
this $ amount of losses
|
|
$500
|
3
|
|
$900
|
13
|
|
$1,000
|
16
|
|
$1,500
|
15
|
|
$1,700
|
15
|
|
$2,000
|
20
|
|
$2,500
|
25
|
|
$2,700
|
20
|
|
$3,000
|
15
|
|
$3,500
|
13
|
|
$3,700
|
12
|
|
$4,000
|
10
|
|
$4,200
|
12
|
|
$4,500
|
8
|
|
$5,000
|
8
|
|
$5,500
|
3
|
a. Calculate the expected value of severity per truck.
b. Calculated the expected loss per truck.
4. The Amazon also has the following data for the number of accidents per truck in the NJ station. The NJ station has 200 trucks. It is known that the variance of frequency per truck in NJ is 0.5519.
|
# of accidents per truck per year
|
# of trucks of having this # of accidents
|
|
0
|
65
|
|
1
|
88
|
|
2
|
47
|
Which truck station is riskier, PA station or NJ station? And why? Show all calculations and explain your numerical results.