Probability of sum of random variables exceed a certain theshold 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:

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*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:

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*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?
 A: 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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