It will be convenient to express a discrete pdf as a continuous pdf using
``delta functions''.
This will make the ensuing discussion easier to follow and simplifies many of
the manipulations for discrete pdf's. Given a discrete pdf 
, let
us associate event 
with the discrete r.v. 
, and then define an
equivalent
``continuous'' pdf as follows:
Here 
is the ``delta'' function and it satisfies the following
properties:
Using these properties, it is straightforward to show that the mean and variance of the equivalent continuous pdf, as defined in Eq. (71), are identical to the mean and variance of the original discrete pdf. Begin with the definition of the mean of the equivalent continuous pdf:
Now take the summation outside the integral and use Eq. (73),
which is the true mean for the discrete pdf. It is left as an exercise to show that this also holds for the variance, and in general for any moment of the distribution.
Much of the material that follows holds for both discrete and continuous pdf's, and this equivalence will be useful in this discussion.
