I have a minor technical issue. Let's say $Y = \sum^{n}_{i=1} X_{i}$. Now I want to find $P(Y > \gamma)$ by Monte Carlo. Let's assume the $X_{i}$ are i.i.d. Gamma distributed. How I see the solution to this problem is the following two cases:
Case 1:
- Generate $n$ random variables: $X_{i} \sim \mathrm{Gamma}(k,\theta)$.
- Check $X_{i} > \gamma$ for each $X_{i}$.
- Take the mean of the result from bullet point 2.
Case 2:
- Generate $m$ random variables $Y \sim \mathrm{Gamma}(n \cdot k,\theta)$.
- Take the mean of the result from the above bullet point.
What would be the right approach?
Recall that Monte Carlo methods are essentially based on the Law of Large numbers. So let's say that you want to approximate:
$$\mathbb{P}(Y > \gamma) $$
Then assuming that you can generate $m$ i.i.d. samples from the distribution of $Y$ you obtain:
$$ \frac{1}{m} \sum_{i=1}^m \mathbb{1}_{\{Y_i > \gamma\}} \stackrel{m \rightarrow \infty}{\rightarrow} \mathbb{E}[\mathbb{1}_{\{Y > \gamma\}}] = \mathbb{P}(Y > \gamma)$$
Where $\mathbb{1}$ is the indicator function. Now, as you pointed out, you are actually able to generate samples $Y_i$ with the method that you proposed or you could just use the following property of the gammas: Sum of independent Gamma distributions is a Gamma distribution. Therefore part 1) of your method is not needed unless you plan to sample the $Y_i$'s as sum of $n$ sampled $X_i$'s. But in general it is not needed to use both part 1) and 2) of your algorithm, part 2) suffices.
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