Let $X_1, X_2$ be independent normal distributions. Consider the quantile $x$ such that $P(X_1 + X_2 \le x) = \alpha$ for some $\alpha \in (0,1)$.

My question is, how does this quantity relate to the quantity $x_1, x_2$ where $x_1$ satisfies $P(X_1 \le x_1) = \alpha$ and $x_2$ satisfies $P(X_2 \le x_2) = \alpha$?

Is $x = x_1 + x_2$? Is it bigger? Is it less? What is the general result for general values of $\alpha$ and specific choices of the distribution parameters?


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