A basic doubt on independence of events in probability

Question

Let $X_1$ and $X_2$ be i.i.d random variable. Now, in a book I see the following steps to calculate $P(X_1 < X_2 < x)$

$P(X_1 < X_2 < x)$

= $P(X_1 < x, X_2 < x, X_1 < X_2)$

Now it is argued that $X_1 < x, X_2 < x$ and $X_1 < X_2$ are independent events. Hence we can factor them. I feel that this is wrong. Am I correct ?

Answer 1

Let $A=[X_1\lt x,X_2\lt x]$ and $B=[X_1\lt X_2]$. If $X_1$ and $X_2$ are i.i.d. and with no atoms then indeed $A$ and $B$ are independent.

To show this, consider $C=[X_2\lt X_1]$ and $D=[X_1=X_2]$ and note that, since $(X_1,X_2)$ and $(X_2,X_1)$ have the same distribution, $$ P[B]=P[C],\qquad P[A\cap B]=P[A\cap C]. $$ Since $(B,C,D)$ is a partition, $2P[B]=1-P[D]$ and $2P[A\cap B]=P[A]-P[A\cap D]$. Thus, $A$ and $B$ are independent if and only if $P[A\cap D]=P[A]P[D]$. If the common distribution of $X_1$ and $X_2$ has no atom, $P[D]=0$ hence the assertion holds..