I have some confusion when understanding the concept of conditional probability.
Given any two random variables $X$ and $Y$ and any two constants $m$ and $k$, Is it true that $$P(Y-X=m | Y > X) = P(Y = m+k | Y > k)? $$
My guess is it is not always true, because
$$\begin{align*} P(Y-X=m | Y > X) & = \sum_{k} P(Y-X=m, X=k | Y > X) \\ & = \sum_{k} P(Y-X=m | X=k, Y > X) P(X=k | Y > X) \\ & = \sum_{k} P(Y-k=m | Y > k) P(X=k | Y > X). \end{align*}$$What if $X$ and $Y$ are i.i.d.? Even further, how about when $X$ and $Y$ are i.i.d. with some memoryless distribution, i.e. exponential or geometric distribution?
In my previous post under the setting of $X$ and $Y$ being i.i.d. with geometric distribution, Henry wrote $$P(Y-X=m | Y > X) = P(Y = m+k | Y > k),$$ about which he said in his comment it is true because "$k$ is merely shorthand for" $X$. But I still don't quite understand that yet.
- If Y is a random variable subject to a memoryless distribution, i.e. exponential or geometric distribution and X is any other random variable with any distribution, is it true that $$P(Y>X+m | Y > X) = P(Y > m)$$
Thanks for your help!