WEBVTT
00:00:00.460 --> 00:00:03.920
We have previously defined
the abstract conditional
00:00:03.920 --> 00:00:06.820
expectation of one random
variable given
00:00:06.820 --> 00:00:08.430
another random variable.
00:00:08.430 --> 00:00:12.300
And we discussed that it is, by
itself, a random variable.
00:00:12.300 --> 00:00:14.880
In particular, it has
an expectation, or
00:00:14.880 --> 00:00:16.690
mean, of its own.
00:00:16.690 --> 00:00:18.560
What is this mean?
00:00:18.560 --> 00:00:20.920
This is what we want
to find out.
00:00:20.920 --> 00:00:24.440
Let us recall our development.
00:00:24.440 --> 00:00:26.970
We look at the conditional
expectation of a random
00:00:26.970 --> 00:00:29.910
variable given a specific
numerical value of another
00:00:29.910 --> 00:00:30.870
random variable.
00:00:30.870 --> 00:00:33.560
This is a number that
depends on little y.
00:00:33.560 --> 00:00:37.430
And this can be used to define
a function little g.
00:00:37.430 --> 00:00:40.610
The function little g for any
particular little y tells us
00:00:40.610 --> 00:00:43.710
the numerical value of the
conditional expectation.
00:00:43.710 --> 00:00:47.690
Since little g is a well defined
function, we can also
00:00:47.690 --> 00:00:51.230
now define this particular
function, which is now a
00:00:51.230 --> 00:00:53.900
function of a random variable.
00:00:53.900 --> 00:00:55.170
It's a well defined object.
00:00:55.170 --> 00:00:56.570
It's a random variable.
00:00:56.570 --> 00:01:00.030
And then we introduced this
abstract notation.
00:01:00.030 --> 00:01:05.140
We defined this object
to be exactly this
00:01:05.140 --> 00:01:07.750
particular random variable.
00:01:07.750 --> 00:01:10.920
So now we want to calculate
the expected value of this
00:01:10.920 --> 00:01:14.440
object, which is written
this way.
00:01:14.440 --> 00:01:18.670
Now this notation, here, may
look quite formidable, but
00:01:18.670 --> 00:01:20.110
let's see what is happening.
00:01:20.110 --> 00:01:23.750
Inside here, we have
a random variable.
00:01:23.750 --> 00:01:27.240
And we take the expected value
of that random variable.
00:01:27.240 --> 00:01:31.510
Or, more crisply, think of that
as the expected value of
00:01:31.510 --> 00:01:37.130
g of capital Y, where g of
capital Y is defined through
00:01:37.130 --> 00:01:39.509
these correspondences here.
00:01:39.509 --> 00:01:42.039
How do we calculate the expected
value of a function
00:01:42.039 --> 00:01:43.560
of a random variable?
00:01:43.560 --> 00:01:46.340
Here we use the Expected
Value Rule.
00:01:46.340 --> 00:01:49.950
Assuming that Y is a discrete
random variable, the Expected
00:01:49.950 --> 00:01:51.775
Value Rule takes this form.
00:01:59.690 --> 00:02:04.590
And the next step is to
substitute the particular form
00:02:04.590 --> 00:02:07.060
for g of Y that we have.
00:02:07.060 --> 00:02:09.919
g of Y was defined
in this manner.
00:02:09.919 --> 00:02:15.570
So we're dealing with the sum
over all little y's of the
00:02:15.570 --> 00:02:19.930
expected value of X, given that
Y takes the value little
00:02:19.930 --> 00:02:26.460
y, weighted by the
PMF of little y.
00:02:26.460 --> 00:02:29.800
Now if we look at this
expression, then it should
00:02:29.800 --> 00:02:31.730
look familiar.
00:02:31.730 --> 00:02:34.480
It is the expression that
appears in the Total
00:02:34.480 --> 00:02:36.380
Expectation Theorem.
00:02:36.380 --> 00:02:39.090
We take the conditional
expectation under different
00:02:39.090 --> 00:02:42.300
scenarios and weigh those
conditional expectations
00:02:42.300 --> 00:02:45.130
according to the probabilities
of those scenarios.
00:02:45.130 --> 00:02:50.000
And this just gives us the
overall expectation of the
00:02:50.000 --> 00:02:52.460
random variable X.
00:02:52.460 --> 00:02:57.150
So this step, here, was carried
out using the Total
00:02:57.150 --> 00:02:58.410
Expectation Theorem.
00:03:04.130 --> 00:03:08.720
So we have proved this important
fact, that the
00:03:08.720 --> 00:03:12.510
expectation of a conditional
expectation is the same as the
00:03:12.510 --> 00:03:14.440
unconditional expectation.
00:03:14.440 --> 00:03:17.610
This important fact is called
the Law of Iterated
00:03:17.610 --> 00:03:18.950
Expectations.
00:03:18.950 --> 00:03:23.030
The proof was carried out
assuming that Y is discrete.
00:03:23.030 --> 00:03:26.820
So we use this particular
version involving a PMF, but
00:03:26.820 --> 00:03:29.320
the proof is exactly the same
for the continuous case.
00:03:29.320 --> 00:03:33.930
You would be using an integral
and the PDF, instead the PMF.
00:03:33.930 --> 00:03:38.250
As the proof indicates, the Law
of Iterated Expectations
00:03:38.250 --> 00:03:42.470
is nothing but an abstract
version of the Total
00:03:42.470 --> 00:03:43.940
Expectation Theorem.
00:03:43.940 --> 00:03:47.380
It is really the Total
Expectation Theorem written in
00:03:47.380 --> 00:03:49.410
more abstract notation.
00:03:49.410 --> 00:03:53.100
But this turns out to be
powerful and also we avoid
00:03:53.100 --> 00:03:55.680
having to deal separately
with discrete or
00:03:55.680 --> 00:03:56.930
continuous random variables.