Let's say there are 100 independent and identically distributed random variables, $Z_i$ for $ 1 \leq i \leq 100$ and 200 independent and identically distributed random variables, $Y_j$ for $1 \leq j \leq 200$. That is, each $Z_i$ and $Y_j$ are also independent of each other.
I want to use the Central Limit Theorem here to find the 95% percentile of the sum of all these random variables. Since the Central Limit Theorem applies when there are a large number of I.I.D random variables, would it be correct to let $X_i = 2Y + Z$ so that S is just the sum from 1 to 100 of these $X_i$'s? For a concrete example:
A portfolio of policies, all issued at the same time, consisting of the following:
- 200 whole life annuity contracts all issued to (50), each with a payment of \$50 annually
- 100 whole life insurance contracts all issued to (40), each with a death benefit of \$10, payable at the moment of death
The future lifetimes of all policyholders are all independent.
You could just say, the sum $S_Z$ of the $Z_i$ is approximately normal with mean $\ldots$ and variance $\ldots$; the sum $S_Y$ of the $Y_j$ is approximately normal with mean $\ldots$ and variance $\ldots$; these two sums are independent, so $S_Z + S_Y$ is approximately normal with mean $\ldots$ and variance $\ldots$.