WEBVTT
00:00:00.980 --> 00:00:04.010
In this segment, we
pursue two themes.
00:00:04.010 --> 00:00:07.220
Every concept has a conditional
counterpart.
00:00:07.220 --> 00:00:10.100
We know about PDFs, but if
we live in a conditional
00:00:10.100 --> 00:00:13.420
universe, then we deal with
conditional probabilities.
00:00:13.420 --> 00:00:16.640
And we need to use
conditional PDFs.
00:00:16.640 --> 00:00:19.500
The second theme is that
discrete formulas have
00:00:19.500 --> 00:00:23.000
continuous counterparts in which
summations get replaced
00:00:23.000 --> 00:00:27.400
by integrals, and
PMFs by PDFs.
00:00:27.400 --> 00:00:30.440
So let us recall the definition
of a conditional
00:00:30.440 --> 00:00:37.010
PMF, which is just the same as
an ordinary PMF but applied to
00:00:37.010 --> 00:00:38.950
a conditional universe.
00:00:38.950 --> 00:00:43.140
In the same spirit, we can start
with a PDF, which we can
00:00:43.140 --> 00:00:46.120
interpret, for example, in
terms of probabilities of
00:00:46.120 --> 00:00:47.970
small intervals.
00:00:47.970 --> 00:00:51.290
If we move to a conditional
model in which event A is
00:00:51.290 --> 00:00:54.010
known to have occurred,
probabilities of small
00:00:54.010 --> 00:00:57.920
intervals will then be
determined by a conditional
00:00:57.920 --> 00:01:01.600
PDF, which we denote
in this manner.
00:01:01.600 --> 00:01:04.010
Of course, we need to assume
throughout that the
00:01:04.010 --> 00:01:07.760
probability of the conditioning
event is positive
00:01:07.760 --> 00:01:10.460
so that conditional
probabilities are
00:01:10.460 --> 00:01:12.400
well-defined.
00:01:12.400 --> 00:01:14.710
Let us now push the
analogy further.
00:01:14.710 --> 00:01:18.060
We can use a PMF to calculate
probabilities.
00:01:18.060 --> 00:01:22.320
The probability that X takes [a]
value in a certain set is
00:01:22.320 --> 00:01:26.020
the sum of the probabilities
of all the possible
00:01:26.020 --> 00:01:28.090
values in that set.
00:01:28.090 --> 00:01:30.890
And a similar formula is true
if we're dealing with a
00:01:30.890 --> 00:01:32.550
conditional model.
00:01:32.550 --> 00:01:38.120
Now, in the continuous case, we
use a PDF to calculate the
00:01:38.120 --> 00:01:42.450
probability that X takes values
in a certain set.
00:01:42.450 --> 00:01:48.300
And by analogy, we use a
conditional PDF to calculate
00:01:48.300 --> 00:01:50.509
conditional probabilities.
00:01:50.509 --> 00:01:54.870
We can take this relation here
to be the definition of a
00:01:54.870 --> 00:01:56.840
conditional PDF.
00:01:56.840 --> 00:02:01.800
So a conditional PDF is a
function that allows us to
00:02:01.800 --> 00:02:05.610
calculate probabilities by
integrating this function over
00:02:05.610 --> 00:02:09.228
the event or set of interest.
00:02:09.228 --> 00:02:12.440
Of course, probabilities
need to sum to 1.
00:02:12.440 --> 00:02:14.810
This is true in the
discrete setting.
00:02:14.810 --> 00:02:17.470
And by analogy, it should
also be true in
00:02:17.470 --> 00:02:19.610
the continuous setting.
00:02:19.610 --> 00:02:23.380
This is just an ordinary PDF,
except that it applies to a
00:02:23.380 --> 00:02:27.050
model in which event A is
known to have occurred.
00:02:27.050 --> 00:02:30.430
But it still is a
legitimate PDF.
00:02:30.430 --> 00:02:33.720
It has to be non-negative,
of course.
00:02:33.720 --> 00:02:36.290
But also, it needs to
integrate to 1.
00:02:39.360 --> 00:02:42.600
When we condition on an event
and without any further
00:02:42.600 --> 00:02:45.900
assumption, there's not much we
can say about the form of
00:02:45.900 --> 00:02:47.470
the conditional PDF.
00:02:47.470 --> 00:02:51.540
However, if we condition on an
event of a special kind, that
00:02:51.540 --> 00:02:56.100
X takes values in a certain
set, then we can actually
00:02:56.100 --> 00:02:58.120
write down a formula.
00:02:58.120 --> 00:03:01.860
So let us start with a random
variable X that has a given
00:03:01.860 --> 00:03:04.786
PDF, as in this diagram.
00:03:11.200 --> 00:03:17.400
And suppose that A is a subset
of the real line, for example,
00:03:17.400 --> 00:03:18.725
this subset here.
00:03:21.860 --> 00:03:24.820
What is the form of the
conditional PDF?
00:03:24.820 --> 00:03:27.620
We start with the interpretation
of PDFs and
00:03:27.620 --> 00:03:29.200
conditional PDFs in terms of
00:03:29.200 --> 00:03:31.180
probabilities of small intervals.
00:03:31.180 --> 00:03:34.650
The probability that X lies in
a small interval is equal to
00:03:34.650 --> 00:03:38.140
the value of the PDF somewhere
in that interval times the
00:03:38.140 --> 00:03:39.600
length of the interval.
00:03:39.600 --> 00:03:42.070
And if we're dealing with
conditional probabilities,
00:03:42.070 --> 00:03:45.320
then we use the corresponding
conditional PDF.
00:03:45.320 --> 00:03:49.200
To find the form of the
conditional PDF, we will work
00:03:49.200 --> 00:03:53.720
in terms of the left-hand side
in this equation and try to
00:03:53.720 --> 00:03:55.450
rewrite it.
00:03:55.450 --> 00:03:57.550
Let us distinguish two cases.
00:03:57.550 --> 00:04:03.780
Suppose that little X lies
somewhere out here, and we
00:04:03.780 --> 00:04:07.160
want to evaluate the conditional
PDF at that point.
00:04:07.160 --> 00:04:11.580
So trying to evaluate this
expression, we consider a
00:04:11.580 --> 00:04:17.238
small interval from little
x to little x plus delta.
00:04:19.850 --> 00:04:25.020
And now, let us write the
definition of a conditional
00:04:25.020 --> 00:04:26.370
probability.
00:04:26.370 --> 00:04:30.160
A conditional probability, by
definition, is equal to the
00:04:30.160 --> 00:04:35.130
probability that both events
occur divided by the
00:04:35.130 --> 00:04:37.385
probability of the conditioning
event.
00:04:41.040 --> 00:04:44.640
Now, because the set A and
this little interval are
00:04:44.640 --> 00:04:49.420
disjoint, these two events
cannot occur simultaneously.
00:04:49.420 --> 00:04:52.540
So the numerator here
is going to be 0.
00:04:52.540 --> 00:04:55.470
And this will imply that
the conditional PDF is
00:04:55.470 --> 00:04:58.620
also going to be 0.
00:04:58.620 --> 00:05:00.470
This, of course, makes sense.
00:05:00.470 --> 00:05:06.130
Conditioned on the event that
X took values in this set,
00:05:06.130 --> 00:05:09.750
values of X out here
cannot occur.
00:05:09.750 --> 00:05:13.000
And therefore, the conditional
density out here
00:05:13.000 --> 00:05:14.830
should also be 0.
00:05:14.830 --> 00:05:21.980
So the conditional PDF is 0
outside the set A. And this
00:05:21.980 --> 00:05:25.680
takes care of one case.
00:05:25.680 --> 00:05:31.150
Now, the second case to consider
is when little x lies
00:05:31.150 --> 00:05:36.250
somewhere inside here inside the
set A. And in that case,
00:05:36.250 --> 00:05:41.760
our little interval from little
x to little x plus
00:05:41.760 --> 00:05:45.070
delta might have this form.
00:05:45.070 --> 00:05:48.460
In this case, the intersection
of these two events, that X
00:05:48.460 --> 00:05:51.870
lies in the big set and X lies
in the small set, the
00:05:51.870 --> 00:05:55.040
intersection of these two events
is the event that X
00:05:55.040 --> 00:05:57.190
lies in the small set.
00:05:57.190 --> 00:06:01.530
So the numerator simplifies just
to the probability that
00:06:01.530 --> 00:06:05.380
the random variable X takes
values in the interval from
00:06:05.380 --> 00:06:08.780
little x to little
x plus delta.
00:06:08.780 --> 00:06:12.480
And then we rewrite
the denominator.
00:06:12.480 --> 00:06:16.110
Now, the numerator is just an
ordinary probability that the
00:06:16.110 --> 00:06:19.870
random variable takes values
inside a small interval.
00:06:19.870 --> 00:06:24.830
And by our interpretation of
PDFs, this is approximately
00:06:24.830 --> 00:06:28.040
equal to the PDF evaluated
somewhere in that small
00:06:28.040 --> 00:06:31.310
interval times delta.
00:06:31.310 --> 00:06:35.570
At this point, we notice that
we have deltas on both sides
00:06:35.570 --> 00:06:36.860
of this equation.
00:06:36.860 --> 00:06:41.240
By cancelling this delta with
that delta, we finally end up
00:06:41.240 --> 00:06:45.180
with a relation that the
conditional PDF should be
00:06:45.180 --> 00:06:48.250
equal to this expression
that we have here.
00:06:48.250 --> 00:06:52.810
So to summarize, we have
shown a formula for
00:06:52.810 --> 00:06:53.930
the conditional PDF.
00:06:53.930 --> 00:06:58.680
The conditional PDF is 0 for
those values of X that cannot
00:06:58.680 --> 00:07:03.340
occur given the information that
we are given, namely that
00:07:03.340 --> 00:07:05.410
X takes values at
that interval.
00:07:05.410 --> 00:07:09.880
But inside this interval, the
conditional PDF has a form
00:07:09.880 --> 00:07:13.700
which is proportional to
the unconditional PDF.
00:07:13.700 --> 00:07:16.630
But it is scaled by a
certain constant.
00:07:16.630 --> 00:07:20.260
So in terms of a picture,
we might have
00:07:20.260 --> 00:07:24.040
something like this.
00:07:24.040 --> 00:07:27.830
And so this green diagram
is the form of
00:07:27.830 --> 00:07:29.145
the conditional PDF.
00:07:32.550 --> 00:07:36.250
The particular factor that we
have here in the denominator
00:07:36.250 --> 00:07:40.510
is exactly that factor that is
required, the scaling factor
00:07:40.510 --> 00:07:44.440
that is required so that the
total area under the green
00:07:44.440 --> 00:07:47.930
curve, under the conditional
PDF is equal to 1.
00:07:47.930 --> 00:07:50.610
So we see once more the
familiar theme, that
00:07:50.610 --> 00:07:53.890
conditional probabilities
maintain the same relative
00:07:53.890 --> 00:07:56.620
sizes as the unconditional
probabilities.
00:07:56.620 --> 00:08:00.620
And the same is true for
conditional PMFs or PDFs,
00:08:00.620 --> 00:08:04.290
keeping the same shape as the
unconditional ones, except
00:08:04.290 --> 00:08:07.660
that they are re-scaled so that
the total probability
00:08:07.660 --> 00:08:12.340
under a conditional
PDF is equal to 1.
00:08:12.340 --> 00:08:15.870
We can now continue the same
story and revisit everything
00:08:15.870 --> 00:08:19.360
else that we had done for
discrete random variables.
00:08:19.360 --> 00:08:22.510
For example, we have the
expectation of a discrete
00:08:22.510 --> 00:08:25.630
random variable and the
corresponding conditional
00:08:25.630 --> 00:08:28.990
expectation, which is just the
same kind of object, except
00:08:28.990 --> 00:08:32.130
that we now rely on conditional
probabilities.
00:08:32.130 --> 00:08:35.919
Similarly, we can take the
definition of the expectation
00:08:35.919 --> 00:08:38.890
for the continuous case and
define a conditional
00:08:38.890 --> 00:08:42.140
expectation in the same manner,
except that we now
00:08:42.140 --> 00:08:44.490
rely on the conditional PDF.
00:08:44.490 --> 00:08:49.140
So this formula here is the
definition of the conditional
00:08:49.140 --> 00:08:52.240
expectation of a continuous
random variable given a
00:08:52.240 --> 00:08:54.710
particular event.
00:08:54.710 --> 00:08:57.970
We have a similar situation with
the expected value rule,
00:08:57.970 --> 00:09:01.250
which we have already seen for
discrete random variables in
00:09:01.250 --> 00:09:05.930
both of the unconditional and
in the conditional setting.
00:09:05.930 --> 00:09:08.810
We have a similar formula
for the continuous case.
00:09:08.810 --> 00:09:11.600
And at this point, you can
guess the form that the
00:09:11.600 --> 00:09:12.960
formula will take in the
00:09:12.960 --> 00:09:17.340
continuous conditional setting.
00:09:17.340 --> 00:09:19.880
This is the expected value
rule in the conditional
00:09:19.880 --> 00:09:24.410
setting, and it is proved
exactly the same way as for
00:09:24.410 --> 00:09:28.260
the unconditional continuous
setting, except that here in
00:09:28.260 --> 00:09:31.560
the proof, we need to work with
conditional probabilities
00:09:31.560 --> 00:09:36.540
and conditional PDFs, instead
of the unconditional ones.
00:09:36.540 --> 00:09:41.370
So to summarize, there is
nothing really different when
00:09:41.370 --> 00:09:44.590
we condition on an event in the
continuous case compared
00:09:44.590 --> 00:09:46.400
to the discrete case.
00:09:46.400 --> 00:09:50.360
We just replace summations
with integrations.
00:09:50.360 --> 00:09:52.930
And we replace PMFs by PDFs.