How to calculate the compound distribution for this actuarial science question?
Suppose you work for an insurance company. Your company offers a certain type of insurance policy, which pays the insured party $10,000 in case of a house fire. Suppose that 1,000,000 people have bought this insurance. Further suppose that each of these people, independent of the other policy holders, has a 0.0002 probability of having a house fire in the next year.
This year, your company has \$3,000,000 to cover these policies. If its total payments exceed this amount, they must take out a loan to cover the difference. Let random variable $Y$ be the size of the loan your company takes out at the end of the year.
(a) We can write $Y$ as a function of another random variable $X$, such that $X$ has a distribution. Give the distribution of $X$ (including all relevant parameters), and write $Y$ as a function of $X$.
(b) What is the probability that your company must take out a loan of at least $1,000,000?
Solution 1:
Let $W$ be the number of fires. Then the technical distribution of $W$ is Binomial, with parameters $n=10^6$ and $p=2\times 10^{-4}$. The total payments $X$ are $10^4W$. The random variable $X$ is well-approximated by the random variable $X'$ which is normally distributed, mean $10^4np$, that is, $2\times 10^6$, and variance $(10^4)^2np(1-p)$, which for all practical purposes is $2\times 10^{10}$.
The random variable $Y$ is $0$ if $X\le 3\times 10^6$, and $X-3\times 10^{6}$ if $X\gt 3\times 10^6$.
As to the probability that $Y\ge 10^6$, we want the probability that $X\ge 4\times 10^6$. The mean of $X$ is $2\times 10^6$, and the standard deviation is about $1.4\times 10^5$. The number $4\times 10^6$ is many many standard deviation units away from the mean, so the probability that $X\ge 4\times 10^6$ is virtually $0$. There is no practical reason to give an estimate, since uncertainties in the model would make such an estimate meaningless.
Remark: I do not work for an insurance company, and find it hard to suppose that I do.