Calculating expected value of unknown random variable 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.

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.
