In this segment we revisit the concept of conditional
expectation and view it as an abstract object
of a special kind.
To get going, let us start with something simple, the
concept of a function.
Let's say a function h that maps real
numbers to real numbers.
As a concrete instance, consider the quadratic
function that maps a number x to its square.
Consider now a random variable, capital X. What do
we mean when we write h of X?
For h defined--
for example in this particular way as a quadratic function--
h of X is defined to be a random variable.
Which random variable?
It is the random variable that takes the value little x
squared whenever capital X, the random variable, happens
to take the value little x.
And this is the random variable that we usually
denote as the random variable X squared.
Now let this come to conditional expectations.
The conditional expectation of a discrete random variable is
defined by this formula.
It is like the ordinary expectation except that we now
live in a conditional universe in which the random variable
capital Y is known to have taken a value little y.
And therefore, instead of using the ordinary formula for
expectations that involve the PMF of X, we now use that
formula but with the conditional PMF of X, which is
the appropriate PMF that applies to
this conditional universe.
And if it happens that the random variable capital X is
continuous, we would have an alternative formula but of the
same kind, where the summation is replaced by an integral and
the PMF is replaced by a PDF.
Now let us look at this quantity here.
We have fixed some particular little y.
Calculate this quantity.
And what we get is a number.
It is a number, but the value of that number depends on the
choice of little y.
If I give you a different little y then you will get
another number for this conditional expectation.
This means that this quantity here is really a
function of little y.
And let us give a name to this function.
Let us call this function g.
Now that we have defined g we can ask, what is this object?
It's a function of capital Y. It's a
function of a random variable.
So it should be a random variable by itself.
By analogy, with the earlier concrete example, it is the
random variable that takes the numerical value g of little y
whenever capital Y happens to take the value little y.
But g of little y has been defined to be the same as this
conditional expectation.
So it's the random variable whose value is this
conditional expectation, which is a particular number, if
capital y happens to take the value little y.
This particular random variable that we have defined
here, g of capital Y, we call it the abstract conditional
expectation of the random variable X, given the random
variable Y.
To summarize, this notation here stands
for a random variable.
It is the random variable whose numerical value turns
out to be this one if the value of the random variable
capital Y happens to be little y.
It is a function of capital Y. Once we know the value of
capital Y, then the value of the conditional expectation is
well defined.
It is known.
And it's equal to this particular number.
It is of course a random variable.
And as a random variable, it has all the attributes that
random variables have.
For example, it has a distribution, that
is, a PMF or a PDF.
It has a mean of its own.
And it has a variance of its own.
So what will be next in our agenda is to talk about these
attributes of this special random variable, and also to
use it in several examples.