= Solution
{c}
For each $i$, the collection of events $\{X_i\leq x\}$, together with the empty event, is a <pi-system>. Its generated <sigma-algebra> is $\sigma(X_i)$, because the half-lines generate the <Borel sigma-algebra> and inverse images preserve intersections and the sigma-algebra operations.
The displayed probability identity says precisely that these two pi-systems are independent. Part (ii) makes $\sigma(X_1)$ and $\sigma(X_2)$ independent, which is the definition of independent <random variables>. Conversely independence of the random variables applies to the half-line events and gives the identity. Thus \b[the joint cumulative distribution function factors exactly when the variables are independent].
Back to article page
