Q:

Probability involving dependent normal variables

I have two independent, identically distributed normal random variables $X \sim N(0,\sigma^2)$, $Y \sim N(0,\sigma^2)$.  I want to know
$$Pr[X-Y>4\sigma \text{ and } X<3\sigma \text { and } Y<3\sigma]$$
Of course, the condition $Y<3\sigma$ seems to be redundant, since we must have $Y<-\sigma$ in order to have $3\sigma>X>4\sigma+Y$.  I also know the fact that $X-Y$ is itself a normal random variable with $X-Y \sim N(0,2\sigma^2)$.  However, $X-Y$ and $X$ are not independent, so I don't know how to separate out the "and"s in the probability condition...

A:

Let $g(x)=\frac1{\sqrt{2\pi}}\mathrm e^{-x^2/2}$, then the probability $p$ you are interested in is
$$
p=\int_{-\infty}^3g(x)\int_{-\infty}^{x-4}g(y)\mathrm dy\,\mathrm dx=\frac1{2\pi}\int_{-\infty}^3\int_{-\infty}^{x-4}\mathrm e^{-(x^2+y^2)/2}\mathrm dy\,\mathrm dx.
$$
Obviously, $\Phi(1)\Phi(3)\leqslant p\leqslant\Phi(3)$ but I see no reason why $p$ should have an analytic expression.

