I have a random variable $Y$ defined as a sum of identical Exponential random variables. The number of those Exponential random variables is Poisson. So $Y$ has finite mean and variance.
I have simulated $2000$ values of $Y$.
I am now estimating $P[Y<3000]$.
Which would give me a better estimation: a normal approximation or just using the simulated values?
In other words, I'm thinking of
Normal Approximation: $$P[Z<\frac{(3000 - \text{ sample mean })}{(\text{ sample standard deviation })}]$$
Simulated values: $\dfrac{\text{sum}(Y < 3000)}{2000}$ (summing over the 2000 simulated values)
I thought that increasing the number of simulated values would be best, so that $Y$ gets closer to a normal distribution by the Central Limit Theorem. But I cannot concretely say why that would be better.
Edit: Given that we have a Poisson and Exponential we can compute the real value of the mean and standard deviation. Perhaps it is these values that should go into the normal approximation instead of the sample mean and variance.