To show this in the random variable picture we need to consider the
new concept of joint probability distribution. If we had an
experiment where each measurement yielded two values 
and
, we would define the joint probability distribution 
of
the random variables X and Y so that the probability of X being
in the range 
and Y being the range
is just 
.
We derive the individual probability distributions 
(or
) from 
through the following argument.
is the probability of finding X in the range
and Y with any value. This
probability is just 
, and so we
may take 
. A similar argument gives
.