# P(X+Y<a | X<a) for X, Y normal and independent distributions

let $$a\in \mathbb{R}$$ and let X and Y be independent normally distributed random variables, with mean $$0$$ and respective variances $$\sigma^2_X$$ and $$\sigma^2_Y$$. Can we express $$P(X+Y with a simple formula (say elementary functions + erf)?

This question is related to this more general one, which hasn't been answered.

• Can you please clarify if the mean is the letter "o" or the number "0"? Mar 14, 2019 at 10:44
• Just a small remark on your terminology: I think you mean that X and Y are normally distributed random variables. Mar 14, 2019 at 11:29
• Thanks, I will correct. Mar 14, 2019 at 12:06