Why $P(X=Y)=0$ if $X$ and $Y$ are i.i.d continous r.v.s? I read this conclusion in text book:
If $X$ and $Y$ are i.i.d continuous r.v.s, then $P(X=Y)=0$, but it doesn't give any proof. I am a bit confused here, what is the rational for this conclusion? Is it because for a continuous r.v., the probability of it equaling to any particular value is $0$?
Thanks.

Try this simple reasoning: Let $Z=X-Y$ (which must also be continuous). Now, \begin{align*}P(X-Y=0)&=P(0 \leq X-Y \leq 0)\\
&= \int_0^0 p(z)dz\\
&=0,
\end{align*}
where $p$ is the density for $Z$.
