Question

In: Statistics and Probability

Lyft has recently launched “Lyft Scooters” in various cities across the U.S. Imagine the company selects...

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.

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

  1. 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?
  1. What is the combined probability distribution when the two loss potentials are pooled?
  1. What is the expected total loss for the distribution of pooled losses?
  1. What is the expected loss per person from the distribution of pooled losses?
  1. What is the standard deviation of the distribution of pooled losses?
  1. What is the standard deviation of the distribution of pooled losses, per person?
  1. 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?

Solutions

Expert Solution

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

Given: Both Zahra and Mateo follow the same distribution as shown below,

Total loss Probability
12000 0.02
0 0.98

The losses are uncorrelated for two distribution. Now, the combined probability distribution is obtained as follows,

Total loss Probability
No accident 0 0.98*0.98=0.9604
Zahra 6000 0.02*0.98=0.0196
Mateo 6000 0.02*0.98=0.0196
Both 12000 0.02*0.02=0.0004

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

The expected loss is obtained using the following formula,


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

The expected loss per person is,


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

The standard deviation for the pooled loss is obtained using the following formula,


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

The standard deviation for the loss per person is,

The expected total loss

Since the expected loss for pooled distribution doesn't depend on the number of people, the expected total loss will be $240

The expected loss per person

The expected total loss will be $240

The standard deviation of the total loss

Since we saw that the standard deviation for the pooled distribution decreases as the number of people increases, the standard deviation will decrease.

The standard deviation of loss per person
Based on your work so far, why might Zahra and Mateo choose to pool their losses?

Since the standard deviation is decreased, they should choose to pool their losses.

orchestra answered 1 year ago
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