WEBVTT

00:00:00.510 --> 00:00:03.170
We now continue with the
development of continuous

00:00:03.170 --> 00:00:06.470
analogs of everything we know
for the discrete case.

00:00:06.470 --> 00:00:08.850
We have already seen a few
versions of the total

00:00:08.850 --> 00:00:12.890
probability theorem, one version
for events and one

00:00:12.890 --> 00:00:14.260
version for PMFs.

00:00:14.260 --> 00:00:16.970
Let us now develop a
continuous analog.

00:00:16.970 --> 00:00:20.230
Suppose, as always, that we have
a partition of the sample

00:00:20.230 --> 00:00:22.740
space into a number of
disjoint scenarios.

00:00:22.740 --> 00:00:24.680
Three scenarios in
this picture.

00:00:24.680 --> 00:00:29.180
More generally, n scenarios
in these formulas.

00:00:29.180 --> 00:00:34.640
Let X be a continuous random
variable and let us take B to

00:00:34.640 --> 00:00:39.150
be the event that the random
variable takes a value less

00:00:39.150 --> 00:00:42.380
than or equal to
some little x.

00:00:42.380 --> 00:00:44.690
By the total probability
theorem, this is the

00:00:44.690 --> 00:00:48.480
probability of the first
scenario times the conditional

00:00:48.480 --> 00:00:52.970
probability of this event given
that the first scenario

00:00:52.970 --> 00:00:56.640
has materialized, and then we
have similar terms for the

00:00:56.640 --> 00:00:58.810
other scenarios.

00:00:58.810 --> 00:01:03.970
Let us now turn this equation
into CDF notation.

00:01:03.970 --> 00:01:09.170
The left-hand side is what we
have defined as the CDF of the

00:01:09.170 --> 00:01:11.370
random variable x.

00:01:11.370 --> 00:01:15.580
On the right-hand side, what we
have is the probability of

00:01:15.580 --> 00:01:20.560
the first scenario multiplied,
again, by a CDF of the random

00:01:20.560 --> 00:01:24.789
variable X. But it is a CDF that
applies in a conditional

00:01:24.789 --> 00:01:29.000
model where event
A1 has occurred.

00:01:29.000 --> 00:01:33.076
And so we use this notation to
denote the conditional CDF,

00:01:33.076 --> 00:01:36.150
the CDF that applies to the
conditional universe.

00:01:36.150 --> 00:01:39.860
And then we have similar terms
for the other scenarios.

00:01:39.860 --> 00:01:43.800
Now, we know that the derivative
of a CDF is a PDF.

00:01:43.800 --> 00:01:47.680
We also know that any general
fact, such as this one that

00:01:47.680 --> 00:01:50.979
applies to unconditional models
will also apply without

00:01:50.979 --> 00:01:53.940
change to a conditional model,
because a conditional model is

00:01:53.940 --> 00:01:57.990
just like any other ordinary
probability model.

00:01:57.990 --> 00:02:00.150
So let us now take derivatives
of both

00:02:00.150 --> 00:02:01.990
sides of this equation.

00:02:01.990 --> 00:02:04.290
On the left-hand side, we
have the derivative of a

00:02:04.290 --> 00:02:07.060
CDF, which is a PDF.

00:02:07.060 --> 00:02:09.570
And on the right-hand side, we
have the probability of the

00:02:09.570 --> 00:02:12.850
first scenario, and then the
derivative of the conditional

00:02:12.850 --> 00:02:18.130
CDF, which has to be the same
as the conditional PDF.

00:02:18.130 --> 00:02:22.690
So we use here the fact that
derivatives of CDFs are PDFs,

00:02:22.690 --> 00:02:27.120
and then we have similar terms
under the different scenarios.

00:02:27.120 --> 00:02:30.430
So we now have a relation
between densities.

00:02:30.430 --> 00:02:33.430
To interpret this relation,
we think as follows.

00:02:33.430 --> 00:02:36.630
The probability of falling
inside the little interval

00:02:36.630 --> 00:02:41.079
around x is determined by the
probability of falling inside

00:02:41.079 --> 00:02:44.670
that little interval under
each one of the different

00:02:44.670 --> 00:02:49.040
scenarios and where each
scenario is weighted by the

00:02:49.040 --> 00:02:52.030
corresponding probability.

00:02:52.030 --> 00:02:58.270
Now, we multiply both sides of
this equation by x, and then

00:02:58.270 --> 00:03:00.690
integrate over all x's.

00:03:03.930 --> 00:03:06.050
We do this on the
left-hand side.

00:03:06.050 --> 00:03:10.340
And similarly, on the right-hand
side to obtain a

00:03:10.340 --> 00:03:12.830
term of this form.

00:03:16.420 --> 00:03:19.520
And we have similar terms
corresponding

00:03:19.520 --> 00:03:21.930
to the other scenarios.

00:03:21.930 --> 00:03:23.470
What do we have here?

00:03:23.470 --> 00:03:27.300
On the left-hand side, we have
the expected value of x.

00:03:27.300 --> 00:03:29.770
On the right-hand side, we
have this probability

00:03:29.770 --> 00:03:35.020
multiplied by the conditional
expectation of X given that

00:03:35.020 --> 00:03:37.350
scenario A1 has occurred.

00:03:37.350 --> 00:03:42.880
And so we obtain a version of
the total expectation theorem.

00:03:42.880 --> 00:03:45.750
It's exactly the same formula
as we had in the discrete

00:03:45.750 --> 00:03:50.650
case, except that now X is a
continuous random variable.

00:03:50.650 --> 00:03:54.300
Let us now look at a simple
example that involves a model

00:03:54.300 --> 00:03:56.530
with different scenarios.

00:03:56.530 --> 00:03:58.800
Bill wakes up in the morning
and wants to go to the

00:03:58.800 --> 00:04:00.190
supermarket.

00:04:00.190 --> 00:04:02.000
There are two scenarios.

00:04:02.000 --> 00:04:06.540
With probability one third,
a first scenario occurs.

00:04:06.540 --> 00:04:11.650
And under that scenario, Bill
will go at a time that's

00:04:11.650 --> 00:04:16.670
uniformly distributed between
0 and 2 hours from now.

00:04:16.670 --> 00:04:22.710
So the conditional PDF of X,
in this case, is uniform on

00:04:22.710 --> 00:04:26.160
the interval from 0 to 2.

00:04:26.160 --> 00:04:30.890
There's a second scenario that
Bill will take long nap and

00:04:30.890 --> 00:04:32.820
will go later in the day.

00:04:32.820 --> 00:04:37.540
That scenario has a probability
of 2/3.

00:04:37.540 --> 00:04:43.340
And under that case, the
conditional PDF of X is going

00:04:43.340 --> 00:04:50.600
to be uniform on the range
between 6 and 8.

00:04:50.600 --> 00:04:53.760
By the total probability theorem
for densities, the

00:04:53.760 --> 00:04:57.250
density of X, of the
random variable--

00:04:57.250 --> 00:04:59.900
the time at which he goes
to the supermarket--

00:04:59.900 --> 00:05:01.840
consists of two pieces.

00:05:01.840 --> 00:05:05.490
One piece is a uniform
between 0 and 2.

00:05:05.490 --> 00:05:11.230
This uniform ordinarily would
have a height or 1/2.

00:05:11.230 --> 00:05:14.230
On the other hand, it gets
weighted by the corresponding

00:05:14.230 --> 00:05:16.670
probability, which is 1/3.

00:05:16.670 --> 00:05:21.966
So we obtain a piece here that
has a height of 1/6.

00:05:21.966 --> 00:05:24.740
Under the alternative scenario,
the conditional

00:05:24.740 --> 00:05:28.770
density is a uniform on the
interval between 6 and 8.

00:05:28.770 --> 00:05:33.870
This uniform has a height of
1/2 again, but it gets

00:05:33.870 --> 00:05:36.480
multiplied by a factor of 2/3.

00:05:36.480 --> 00:05:40.190
And this results in a height
for this term that we have

00:05:40.190 --> 00:05:43.130
here, which is 1/3.

00:05:43.130 --> 00:05:47.810
And this is the form of the PDF
of the time at which Bill

00:05:47.810 --> 00:05:49.076
will go to the supermarket.

00:05:53.810 --> 00:05:57.790
We can now finally use the total
expectation theorem.

00:05:57.790 --> 00:06:00.840
The conditional expectation
under the two scenarios can be

00:06:00.840 --> 00:06:01.720
found as follows.

00:06:01.720 --> 00:06:05.530
Under one scenario, we have
a uniform between 0 and 2.

00:06:05.530 --> 00:06:08.350
And so the conditional
expectation is 1, and it gets

00:06:08.350 --> 00:06:11.210
weighted by the corresponding
probability, which is 1/3.

00:06:11.210 --> 00:06:16.690
Under the second scenario, which
has probability 2/3, the

00:06:16.690 --> 00:06:21.410
conditional expectation is the
midpoint of this uniform,

00:06:21.410 --> 00:06:23.080
which is 7.

00:06:23.080 --> 00:06:26.320
And this gives us the expected
value of the

00:06:26.320 --> 00:06:28.350
time at which he goes.

00:06:28.350 --> 00:06:32.010
So this is a simple example, but
it illustrates nicely how

00:06:32.010 --> 00:06:35.380
we can construct a model
that involves a number

00:06:35.380 --> 00:06:36.870
of different scenarios.

00:06:36.870 --> 00:06:40.180
And by knowing the probability
distribution under each one of

00:06:40.180 --> 00:06:43.210
the scenarios, we can
find the probability

00:06:43.210 --> 00:06:45.600
distribution overall.

00:06:45.600 --> 00:06:49.140
And we can also find the
expected value for the overall

00:06:49.140 --> 00:06:50.390
experiment.