Question

Lyft has recently launched “Lyft Scooters” in various cities across the U.S. Imagine the company selects two individuals, Zahra in Madison, Wisconsin, and Mateo in Albuquerque, New Mexico, to test Lyft Scooters in their respective cities. Each recognizes that she or he has a 2% chance of experiencing an accident. If an accident occurs, a $12,000 will be lost due to injury and property damage.

1. What is Zahra’s probability distribution for loss arising out of an accident?

2. What is Zahra’s expected value of loss from an accident?

3. What is the standard deviation of Zahra’s probability distribution of loss arising out of an accident?

For questions 4-11, assume that Zahra and Mateo decide to pool (or share equally) their losses. The losses are uncorrelated.

4. How much will each person need to pay into the pot of pooled money in order to cover their losses in the event they both experience an accident?

5. What is the combined probability distribution when the two loss potentials are pooled?

6. What is the expected total loss for the distribution of pooled losses?

7. What is the expected loss per person from the distribution of pooled losses?

8. What is the standard deviation of the distribution of pooled losses?

9. What is the standard deviation of the distribution of pooled losses, per person?

10. If the number of people in the pool increases from 2 to 1000, how will this change influence each of the following? Use words, not numbers, to state whether they will they increase, decrease or stay the same.
    
    1. The expected total loss
       
       
    2. The expected loss per person
       
       
    3. The standard deviation of the total loss
       
       
    4. The standard deviation of loss per person

Based on your work so far, why might Zahra and Mateo choose to pool their losses?

Answer 1

Let the event of loss be denoted as X. In case of no loss X = 0 and in event of loss X = 1.

The porbability distribution of Zahra's loss is:

P(X=0) = 0.98

P(X=1) = 0.02

The expected value of loss from the accident is:

E(X) = 0.98*0 + 0.02*12000 = $240

E(X^2) = 0.98*0^2 + 0.02*12000^2 = 2880000

V(X) = E(X^2) - E(X)^2 = 2880000 - 240^2 = 2822400

sigma(X) = 1680

Now we have that Zahra and Mateo have independent and identical losses.

The expected loss is:

P(Y=0) = 0.98*0.98 = 0.9604

P(Y=1) = 0.98*0.02*2 = 0.0392

P(Y=2) = 0.02*0.02 = 0.0004

The expected loss is:

E(Y) = 0.9604*0 + 0.0392*12000 + 0.0004*24000 = $480

which is nothing but the sum of the individual expected losses.

In the event that both face an accident the amount of money required is 0.0004*24000 = $9.6

The value each of them has to pay is 9.6/2 = $4.8

The standard deviation of the pooled losses is:

E(Y^2) = 0.9604*0^2 + 0.0392*12^2+ 0.0004*24000^2 = 5875200

V(Y) = 5644800

sigma(Y) = $2375.88

The standard deviation of pooled losses per person is:

2375.88/2 = $ 1187.94

If the number of people increases from 2 to 1000 then

a) the expected total loss will increase by the number of people.

b) Expected total loss per person will remain the same.

c) The standard deviation fo the total loss will increase.

d) The standard deviation of loss per person will decrease with the increase in the number of people.

Zahra and Mateo will pool their losses since there is no change in the expected loss that they will suffer in either of the cases. However, the standard deviation of the individual losses decreases with the pooling of the losses with beneficial for both of them.