Moving on to the next chapter “Random Variables – I”, take a look at the following problem. Show that the random variable is continuous if and only if
for all
.
(Forward direction) Suppose is a continuous random variable, then its distribution function
is also continuous by definition. Hence,
by definition of continuity.
(Reverse direction) Suppose for all
and let
be the corresponding distribution function. Let
be a sequence of sets such that
such that
. Then,
because
is countably additive. Thus,
and since
we see that
for any
which is the definition of continuity.