Expected value of ratio of normal CDFs

Question

I am trying to compute the expected value of the ratio of two normal CDFs. Specifically, I like to compute the expected value of $\Phi(X+Y)/\Phi(X)$ where $X$ and $Y$ are independent normally distributed variables with means $\mu_X$ and $\mu_Y$ and variances $\sigma_X^2$ and $\sigma_Y^2$, and where $\Phi(.)$ is the standard normal CDF function.

I am not sure whether there is a closed form solution. In case there isn't any approximation to the derivative of the expected value in $\mu_X$ would be very helpful for my problem, too.

I would appreciate any help or suggestions where to look further.

Answer 1

This is too much for a comment, but here's a simulation, for $X \sim \mathcal{N}(1, 4)$ and $Y \sim \mathcal{N}(2, 16)$.

ratio_sim <- function(nsims, nnorms, mu_X, mu_Y, sd_X, sd_Y, seed=30){
set.seed(seed)
means <- vector()
for (i in 1:nsims){
X <- rnorm(nnorms, mean = mu_X, sd = sd_X)
Y <- rnorm(nnorms, mean = mu_Y, sd = sd_Y)
sum <- X+Y
cdf_sum <- pnorm(sum)
cdf_X <- pnorm(X)
ratio <- cdf_sum/cdf_X
means <- c(means, mean(ratio))
}
return(means)
}

#For example:
ratio_sim(nsims = 5000, nnorms = 5000, mu_X = 1, mu_Y = 2, sd_X = 2, sd_Y = 4)