Say that we have three i.i.d random variables $X,Y,Z$. Each has pdf $f(\cdot)$ and cdf $F(\cdot)$, and furthermore, the difference of any two (e.g. $Y-X$) has pdf $f_d(\cdot)$ and cdf $F_d(\cdot)$.
The problem is to calculate the probability of these two events occurring: $Pr(Y-X<c\cap Z-X<d)$ for known $c,d$.
So, we want: $Pr(Y-X<c)\cdot Pr(Z-X<d|Y-X<c)$
At the moment, I'm specifically working on $X\sim N(0,1)$, so $Y-X \sim N(0,2)$, but knowing the general answer would be useful.
- Does this calculation depend on knowing whether $c>d$?
- What's the answer? :)
Any help would be greatly appreciated!