Tails of a Conditional Normal Distribution

Question

Let $X \sim N (0, \sigma^2)$ and $Y ~ \sim N(0, 1 + \sigma^2)$ be independent. I'm trying to understand and visualize the function $$f(x) := P(X > Y + x | X > x),$$ for large $x$ (say, $x > 3 \sigma$). For example, for $\sigma = 1$, what is $P(X > Y + 5 | X > 5)$? I have run some large simulations and my suspicion is that as $x \to \infty$, $f(x)$ converges to a constant in (0, 1), but I'm stuck as to how to evaluate, approximate, or visualize this function.

Answer 1

For your example if $X\sim \mathcal N(0,1)$ and $Y\sim N(0, 2)$ then $$\begin{align}P(X>Y+5|X>5)&=P(X>Y+5,Y>0|X>5)+P(X>Y+5,Y\le0|X>5)\\ &=P(X>Y+5|X>5,Y>0)P(Y>0)+P(X>Y+5|X>5,Y\le0)P(Y\le0)\\ &=\int_0^\infty\int_{Y+5}^\infty \frac{f(x)}{P(X>5)}\frac{g(y)}{P(Y>0)}dxdy\cdot \frac 12+1\cdot \frac 12\end{align}$$