Question

Bespin Car Rental predicts that the annual probability of one of its cars being destroyed in a crash is 1 in 1,000,000. If destroyed, the value of the property damage to the car equals $45,000. Assume that there are no partial losses; the car is either destroyed in a crash or suffers no loss. A) Show the physical damage loss distribution for Bespin Car Rental’s automobiles and calculate the expected value of the physical damage loss. B) Show the calculations for the variance and the standard deviation.

Answer 1

(A) The expected loss is $0.045.

(B) The variance and standard deviation of physical damage loss are $2,025 and $45 respectively.

Explanation:

The annual probability of Bespin Car Rental's cars being destroyed is 1 in a million, i.e 0.000001.

It is assumed that the car is either destroyed or there was no loss suffered.

The loss amount in case the car is destroyed is, $45,000.

(A)

The distribution for physical damage loss is displayed in the table below.

The Expected value of physical damage loss is:

`E(X)=\sum xP(X)=(45000*0.000001)+(0*0.999999)=0.045`

Thus, the expected loss is $0.045.

(B)

The variance of a random variable X is: Var (X) = E (X²) - [E (X)]².

The variance of physical damage loss is:

Compute the variance as follows:

`Var(X)=E(X^(2))-[E(X)]^(2)\\=\sum x^(2)P(X)-[\sum xP(X)]^(2)\\=[(45000^(2)*0.000001)+(0^(2)*0.999999)]-(0.045)^(2)\\=2025-0.002025\\=2024.997975\\\approx2025`

The standard deviation of physical damage loss is:

`SD=√(Var(X))=√(2025)=45`

Thus, the variance and standard deviation of physical damage loss are $2,025 and $45 respectively.