If $X$ and $Y$ are two independent and identically distributed continuous random variables how could I find $P (X < Y)$?

Question

I've thought that it could relate to a joint distribution function of the two however I am unsure of how I would go about using this to determine $P(X < Y)$, how should I approach this problem?

Thanks in advance.

Answer 1

Note that $P(X<Y) + P(Y<X) + P(X=Y)=1$. Since $X, Y$ are continuous, $X-Y$ is continuous, so $P(X-Y=0)=0$. Then by symmetry $P(X<Y)=P(Y<X)$ so $2P(X<Y)=1$ hence the solution is ${1\over 2}$.