# Approximate CDF of the sum of a gaussian and a truncated gaussian

I am looking for a quick-to-compute approximation of the CDF of $X+Y$, where $X \sim N(0,\sigma_1^2)$ and $Y$ is a truncated gaussian, more specifically, a gaussian with mean $0$, standard deviation of $\sigma_2$, truncated on one side, at $a$ ($a < 0$). I have found a closed-form density function for the distribution of $X+Y$ here, but not a CDF.

Unfortunately, using the density function means that I have to numerically integrate it repeatedly, which is computationally very intensive, if done many times. Is there an approximation to the CDF of $X+Y$, which I could use instead?

I'd appreciate any suggestions or pointers on how to solve this problem.

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Does the Gaussian CDF $\Phi$ qualify as a closed-form function for you? Then you could as well define a function $\Psi$ as the CDF of $X+Y$ and call that a closed-form solution. The only difference is that more is known about the former and better algorithms are available for evaluating it. –  Eckhard Mar 28 '13 at 9:58
But how would you go about defining $\Psi$ given that Y is truncated on one side? –  Pavel Mar 28 '13 at 11:22
@Eckhard: I suppose what I am really looking for is not necessarily a closed-form solution, but rather something that's quick to compute. I changed my question accordingly. –  Pavel Mar 29 '13 at 11:17