Gfrktyl

Say I have two random variables X1 and X2 and that they are independent. Am I guaranteed that the events "X1 is less than x1" and "X2 is less than x2" are independent?

If not, under which conditions is this the case? Or better, what is a sufficient condition for having independence between those two events?

I remember that the definition of two random variables $X_1$ and $X_2$ are independent is for any event generated by random variables $X_1$ and event generated by $X_2$ are independent. So events $(X_1<x_1)$ and $(X_2<x_2)$ are independent is the condition that two random variables $X_1$ and $X_2$ are independent.

Maybe you need to check the textbook or search the internet to find the definition of independent of two random variables.

If $X$ and $Y$ are independent random variables, then it is always the case that the events $A = {X leq a}$ and $B = {Y leq b}$ are independent events. Specifically, one of the (equivalent) definitions of independence of two random variables is that the joint CDF factors into the product of the individual (a.k.a. marginal) CDFs. That is, we are insisting that independence of $X$ and $Y$ means that
$$F_{X,Y}(a,b) = F_X(a)F_Y(b)~text{for all real numbers}~a~text{and}~btag{*}$$
But, $$F_{X,Y}(a,b)
stackrel{Delta}{=} Pleft({X leq a, Y leq b}right) = Pleft({Xleq a}cap {Y leq b}right) = P(Acap B)$$
while $$F_{X}(a)
stackrel{Delta}{=} Pleft({X leq a}right) = P(A), quad F_{X}(b)
stackrel{Delta}{=} Pleft({Y leq b}right) = P(B)$$
and so $(*)$ is saying that
$$P(Acap B) = P(A)P(B),$$
that is, $A$ and $B$ are independent events.