Conditional expection problem

[xsimpx]

An insurance company designates 10% of its customers as high risk and 90% as low risk. The number of claims made by a customer in a calendar year is Poisson distributed with mean theta and is independent of the number of claims made by a customer in the previous calendar year. For high risk customers theta=0.6, while for low risk customers theta=0.1. Calculate the expected bumber of claims made in calendar year 1998 by a customer who made one claim in calendar year 1997.



Answer = 0.24

Could someone explain how they arrive at this solution.

[atomic]

Let H be the event that the customer is high risk, and L be the event that the customer is low risk. Then let N(y) be the number of claims made by a given customer in CY y. So

.

This is simply Bayes' theorem. Now using the law of total probability to expand the denominator,

.

Since we are given

,

,

and furthermore, Pr[H] = 0.1, Pr[L] = 0.9, we easily find

.

Then we find

Pr[L | N(97) = 1] = 1 - 0.287929 = 0.712071,

and it follows that

E[N(98) | N(97) = 1] = E[N(98) | H] Pr[H | N(97) = 1] + E[N(98) | L] Pr[L | N(97) = 1].

Since the expected value of a Poisson distribution is equal to its parameter, we see that the above simplifies to

E[N(98) | N(97) = 1] = (0.6)(0.287929) + (0.1)(0.712071) = 0.243964.

[xsimpx]

Thanks that makes a lot more sense now.

[cmichener]

Wait, why is this conditional if the number of claims from year to year is independent?

[Gandalf]

There are many customers. atomic's claims in one year are independent of atomic's claims in another year. xsimpx's claims in one year are independent of xsimpx's claims in another year.

In this problem, you are looking at the same customer in both years. This means the number of claims in the first year influences the probability that this particular customer is high risk, and thus influences the expected number of claims in the second year.

To take a more extreme example, suppose low risk always has 1 or 2 claims, p=.5 each; high risk always has 10 or 20, p=.5 each. If you see 10 claims in year 1, then you know you have a high risk customer. So in year 2, p(10)=p(20)=.5. But the fact that this high risk customer had 10 rather than 20 in year 1 doesn't change his likelihood of 10 vs 20 in year 2.

[atomic]

Originally Posted by cmichener

Wait, why is this conditional if the number of claims from year to year is independent?

Because knowledge of the past claims experience provides partial information about the nature of the insured's risk class, which is not independent from year to year, but is (implied) a fixed property of the insured. Thus the expectation is conditional, since the more prior observations you have, the better your confidence of which risk class the insured belongs to, better than the 10% high, 90% low distribution that only applies in the absence of any information.

A simple example is in the tossing of a die that you do not know whether it is biased or fair. You inspect it closely and prior to any trial, you have no reason to say either way. As you roll the die and record the results you can start to infer with greater confidence whether the die is biased (but you can never truly be certain). This notion of revising one's prior belief based on observed outcomes is the foundation of Bayesian probability theory.