The question:
Micro Insurance Company issued insurance policies to $32$ independent risks. For each policy, the probability of a claim is $1/6$. The benefit amount given that there is a claim has probability density function
$$ f(y) =
\begin{cases}
2(1-y) & 0<y<1,
\\0 & \text{otherwise}.
\end{cases}$$
Calculate the expected value of total benefits paid.
My attempt:
I'm not sure on how to define my random variable. Its expected value should sum from 1 to 32, each with probability $\frac{1}{6} \int f(y) dy$, I think.
Best Answer
Let $Y_{n}$ be the amount paid to the $n$-th policyholder assuming that the claim is made. Let $\mathbb{I}_{n}$ be $0$ when the $n$-th claim is not made and $1$ otherwise. Then, the total benefits paid is $$ X=\sum_{n=1}^{32}\mathbb{I}_{n}Y_{n}. $$ We need to calculate $$ \mathbb{E}\left[X\right]=\mathbb{E}\left[\sum_{n=1}^{32}\mathbb{I}_{n}Y_{n}\right]=\sum_{n=1}^{32}\frac{1}{6}\int_{0}^{1}yf\left(y\right)dy. $$ I think you can do the rest yourself.