Conditional Distribution of $X-Y$ given $X > Y$ when $X$ follows exponential distribution

Question

Suppose $X$ follows exponential distribution with a positive parameter $\lambda$ and $Y$ is a positive continuous random variable, independent of $X$. Then what is the conditional distribution of $X-Y$ given $X > Y$?

Answer 1

Consider the distribution of $X\vert X>Y$ or the memorylessness. Then see if it is easy to change it to $X-Y\vert X>Y$

But here Y is also a random variable. Does it matter?

If you know $\mathbb{P}(A \vert B)$ and $B$ is itself a random variable, then you can find the probability of $\mathbb{P}(A)$ as a compound distribution or by using the law of total probability

$$\mathbb{P}(A) = \sum_{\forall B} \mathbb{P}(A \vert B)\mathbb{P}(B) $$

if $ \mathbb{P}(A \vert B) = f(A)$ is a function independent of $B$ then it can be taken out of the sum and you get

$$\mathbb{P}(A) = \sum_{\forall B} \mathbb{P}(A \vert B)\mathbb{P}(B) = f(A)\sum_{\forall B} \mathbb{P}(B) = f(A) $$

Similarly when $\mathbb{P}(X-Y\vert X>Y, Y)$ is independent from $Y$ then you know $\mathbb{P}(X-Y\vert X>Y)$