Pooling, central tendency and dispersion
Develop an understanding of the consequences of pooling on measurements of central tendency and dispersion under various situations.
Two friends, Anastasia and Junhao, live in different apartment complexes. Each recognizes that he or she has a .01 probability of experiencing a fire. If a fire is experienced in a given person’s apartment, $8,000 in belongings will be lost.
1. What is Junhao’s loss distribution?
2. What is Junhao’s expected loss?
3. What is the standard deviation of Junhao’s loss distribution?
For questions 4-11, assume Anastasia and Junhao decide to pool their losses and that losses are uncorrelated.
4. If both Anastasia and Junhao experience a loss, how much will each person need to pay to replace the damaged belongings?
5. What is the full pooled loss distribution for the two?
6. What is the expected total loss for the pooled distribution?
7. What is the expected loss per person from the pooled distribution?
8. What is the standard deviation of the pooled distribution?
9. What is the standard deviation of the pooled distribution, 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 (describe the effect in words, not numbers)?
a. The expected total loss
b. The expected loss per person
c. The standard deviation of the total loss
d. The standard deviation of loss per person
11. Based on your work so far, why might Anastasia and Junhao choose to pool their losses?
For questions 12-16 assume that Anastasia and Junhao live in the same apartment complex, so that their loss experiences are positively correlated.
12. If we know that Junhao experiences a fire, what can be said about the probability that Anastasia experiences a fire (relative to the independent case)?
13. If we know that Junhao does not experience a fire, what can be said about the probability that Anastasia experiences a fire (relative to the independent case)?
14. What is the effect of positive correlation on the expected loss per person? Specifically, what can be said about the expected loss per person, if the losses are positively correlated, relative to each of the following?
a. The expected loss per person under zero correlation:
b. The expected loss per person without pooling:
15. What is the effect of positive correlation on the standard deviation of loss per person? Specifically, what can be said about the standard deviation of loss per person, if the losses are positively correlated, relative to each of the following?
a. The standard deviation of loss per person under zero correlation:
b. The standard deviation of loss without pooling:
16. Should Anastasia and Junhao still pool their losses when positive correlation exists if their intent is to reduce risk? Why or why not?
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