By this point in this class, you must have realized that a
lot of revolves around the concept of conditioning.
Conditional expectations play a central role.
For this reason, it is useful to revisit this concept and
view it in a more abstract manner.
The basic idea is that the value of a conditional
expectation is affected by a random quantity by the value
of the random variable Y on which we are conditioning.
It is a function of Y and, therefore, a random variable.
Based on this observation, we will redefine the conditional
expectation as a random variable and then try to
understand its properties.
In particular, we will develop a formula for the expected
value of the conditional expectation.
This will be what as known as the law of iterated
expectations.
After doing all this, we will follow a similar program for
the conditional variance.
Once more, we will see that it can be
viewed as a random variable.
And then we will relate its expected value with the
unconditional variance.
This will be the so-called law of total variance.
As an illustration of the tools we are introducing in
this lecture, we will consider various examples that will
hopefully clarify the concepts involved.
Our final and most important example will involve the sum
of a random number of independent random variables.
The setting here is more challenging than the case
where we add a fixed number of random variables.
But by using conditioning, we will be able to derive
formulas for the mean and the variance.