Let $X, Y, Z$ be iid continuous random variables. Show that $P[X < Y ] = 0.5$. Calculate $P[X < Y < Z]$.
My try: We know that for continuous random variable, $P(X=y)=0$, i.e. Probability at a point is zero. Hence, $P(X>y)+P(X<y)=1$. But how to show that $P(X<Y)=P(X>Y)=0.5.$ Can you suggest a book to study this concepts?.