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 considered separately, the numbers of claims are independent Poisson r.v.'s with the same lambda. 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 couple make? Show that the answer to the last question is the same for an arbitrary distribution of the number of claims

Explanation / Answer

Let X be a random variable which denotes the number of claims made on the wife’s policy.

And let Y be a random variable which denotes the number of claims made on the husband’s policy.

It is given in the question that X~P() and Y~P() and also X and Y are independent random variables.

Now, let S=X+Y So, we can say that S~P(+)=P(2)i.e p(s)= e2s/s!

*This is a property of poisson distribution which states that the sum of independent poisson variates is also a poisson variate with parameter equal to the sum of the parameters of each variable in the sum.

Now, when we consider the case of joint insurance, the distribution of the claims becomes the product of the distribution of each random variable(due to property of independent random variable).So, let the variable be Z which denotes the probability of joint claims then the joint probability function is given by

P(z)= (e x/x!)* (e y/y!) = e2x+y/x!y!

Now, the probability of making one claim each in case of individual policy is S=1+1=2

So,P(S=2)= e22/2!

And the probability of making one claim each in the case of joint insurance is

P(z)= e21+1/1!1!= e22=2.P(S=2)

So, from the above we see that the probability of making one claim in the case of joint insurance is twice the probability of making individual claims. It is also given that the premium of joint insurance policy is double the premium of individual policy so the cost to the couple will be the same. But the claims with priority in the case of joint insurance is 2 whereas in the case of individual policy is 1 and also the joint policy covers both the husband and the wife. Hence, we can say that it will better for them to get the joint policy.

Now, no matter what distribution we consider the above argument is going to remain the same. That is the cost will be the same but they will have more risk coverage if they buy the joint probability. The probability of making one claim separately is half the probability of making one claim each together. And the get two priority claims in the case of a joint probability as opposed to only one priority claim in the case of individual policy.

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