Let $f(x,y,z)=e^{-x-y-z},x>0,y>0,z>0$, and $=0$ otherwise, be the joint PDF of $(X,Y,Z)$. Compute $P(X=Y<Z)$.

Question

Let $$f(x,y,z)=\begin{cases}e^{-x-y-z}, \ \ \text{$x>0,y>0,z>0$} \\0,\ \ \text{otherwise}\end{cases}$$ be the joint PDF of $(X,Y,Z)$. Compute $P(X=Y<Z)$.

According to the textbook, the answer is $0$, but I am unsure how to proceed.

Answer 1

It is zero, because for continuous random variables events like “$X = \text{something}$” have zero probability. More precisely, to calculate given probability, you have to integrate joint PDF over region $\{(x, y, z) : x=y, y<z\}$ which has zero measure (volume) in $\mathbb{R}^3$.