Poisson Binomial Distribution Moments

Question

I'm carrying out a piece of work where we are trying to approximate the distribution and parameters of a problem where we assume there are a fixed number of insurance policies y, each of which has a fixed sum assured (money paid on claim). Each person holding a policy has an identical probability of death (q). Therefore what is the distribution of the total claims in a year.

Letting the total claims in a year be T and I be the indicator random variable binomially distributed with n of 1, prob of success (claim) of q, and Si being the sum assured per policy, the total is then:

$T = \sum_{i=1}^{y} I_i S_i$

I know that it will be normally distributed (at least approximates) due to CLT, with:

$E[X]=yq \sum S_i$

But I cannot begin the derivation of the variance of the underlying distribution. I have run monte carlo simulations and compared the variance with what I think it could be (typically binomial).

Any help would be appreciated.

Thanks!

Answer 1

Hints:

• For an individual, the payout is $S_i$ with probability $q$ and $0$ with probability $1-q$
• From this you can find the expected payout for that individual, and the variance of the payout for that individual
• The expected total payout is the sum of the payouts for the individuals
• If the claims are independent then the variance of the total payout is the sum of the variances of the payouts for the individuals