A husband and wife each can purchase insurance for which the payoff for the firs

Question

A husband and wife each can purchase insurance for which the payoff for the first claim is much higher than the others in the same year. In the husband’s and the wife’s cases consid- ered separately, the numbers of claims are independent Poisson r.v.’s with the same . The couple has also an option to buy a joint insurance where the number of claims with priority is two. Find the distribution of the total number of the claims with priority covered for the case of the two separate insurances and for the case of the joint insurance. If the premium for the joint insurance is double the premium of the individual policy, what decision should the cou- ple make? Show that the answer to the last question is the same for an arbitrary distribution of the number of claims (not only Poisson). A husband and wife each can purchase insurance for which the payoff for the first claim is much higher than the others in the same year. In the husband’s and the wife’s cases consid- ered separately, the numbers of claims are independent Poisson r.v.’s with the same . The couple has also an option to buy a joint insurance where the number of claims with priority is two. Find the distribution of the total number of the claims with priority covered for the case of the two separate insurances and for the case of the joint insurance. If the premium for the joint insurance is double the premium of the individual policy, what decision should the cou- ple make? Show that the answer to the last question is the same for an arbitrary distribution of the number of claims (not only Poisson). A husband and wife each can purchase insurance for which the payoff for the first claim is much higher than the others in the same year. In the husband’s and the wife’s cases consid- ered separately, the numbers of claims are independent Poisson r.v.’s with the same . The couple has also an option to buy a joint insurance where the number of claims with priority is two. Find the distribution of the total number of the claims with priority covered for the case of the two separate insurances and for the case of the joint insurance. If the premium for the joint insurance is double the premium of the individual policy, what decision should the cou- ple make? Show that the answer to the last question is the same for an arbitrary distribution of the number of claims (not only Poisson).

Explanation / Answer

Let X be the number of claims made by the husband and Y be the number of claims made by the wife. According to the given question, X ~P() and Y~P()

i.e P(X=x) = e-()x /x!   and P(Y=y) = e-()y /y!

Since, the number of claims made by the husband is independent of the number of claims made by the wife. So, if we consider S=X+Y we can say that S~P(+) i.e S~ P(2).

So, the probability function of S is given as P(S=s) = e-2(2)s /s!

And, the joint probability function is given as P(X=x)*P(Y=y) = e-()x /x! * e-()y /y! = e-2()x+y /y!x!

It is given that the premium for the joint insurance is double the premium of the individual policy,

Let us consider when S=2(i.eX=1 & Y=1), P(S=2) = e-222

And, P(X=1).P(Y=1)= e-2()2

We see that P(S=2)= 2.P(X=1).P(Y=1)

So, we see that the probability of making one claim each in the case of individual policy is twice the probability of making 2 claims in the case of joint policy. Plus we know that the premium of joint insurance is double the premium of the individual policy. Based on the given facts we then conclude that the couple should take individual policies.

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