WEBVTT
00:00:00.330 --> 00:00:03.510
Here is a simple application
of the law of iterated
00:00:03.510 --> 00:00:04.780
expectations.
00:00:04.780 --> 00:00:08.150
We revisit the stick-breaking
example, which we have seen
00:00:08.150 --> 00:00:10.190
sometime in the past.
00:00:10.190 --> 00:00:12.700
So in this example, we start
with a stick that has a
00:00:12.700 --> 00:00:16.340
certain length and which we
break at a point that's chosen
00:00:16.340 --> 00:00:20.980
uniformly at random throughout
the length of the stick.
00:00:20.980 --> 00:00:25.880
And we call the point at which
we cut the stick capital Y.
00:00:25.880 --> 00:00:29.090
So the random variable Y has a
uniform distribution on the
00:00:29.090 --> 00:00:32.320
interval from 0 to l,
and is described by
00:00:32.320 --> 00:00:34.210
this particular PDF.
00:00:34.210 --> 00:00:38.090
Then we take the piece of the
stick that's left and we break
00:00:38.090 --> 00:00:42.390
it at a point that's chosen
uniformly over the length of
00:00:42.390 --> 00:00:43.910
the stick that's left.
00:00:43.910 --> 00:00:49.670
So the stick that was left has
a length Y, and the place at
00:00:49.670 --> 00:00:54.420
which we cut it, X, is chosen
uniformly over that interval.
00:00:54.420 --> 00:00:57.390
So in particular, X-- or
rather the conditional
00:00:57.390 --> 00:00:59.880
distribution of X given Y--
00:00:59.880 --> 00:01:03.150
is uniform on that interval.
00:01:03.150 --> 00:01:06.740
So in this example, what is the
expected value of X if I
00:01:06.740 --> 00:01:08.210
tell you the value of Y?
00:01:08.210 --> 00:01:11.590
Well, given the value of Y,
the random variable X is
00:01:11.590 --> 00:01:12.930
uniform on that range.
00:01:12.930 --> 00:01:16.370
So the expected value is going
to be at the midpoint that is
00:01:16.370 --> 00:01:19.400
equal to y over 2.
00:01:19.400 --> 00:01:22.080
This is an equality
between numbers.
00:01:22.080 --> 00:01:24.480
For any particular
number, little y,
00:01:24.480 --> 00:01:26.680
we have this equality.
00:01:26.680 --> 00:01:31.700
Now let us convert this concrete
equality between
00:01:31.700 --> 00:01:34.600
numbers to a more abstract
equality
00:01:34.600 --> 00:01:36.840
between random variables.
00:01:36.840 --> 00:01:40.910
This object is a random variable
that takes this value
00:01:40.910 --> 00:01:43.410
whenever capital
Y is little y.
00:01:43.410 --> 00:01:47.580
So this is an object that takes
the value little y over
00:01:47.580 --> 00:01:51.509
2 whenever the random variable
capital Y happens
00:01:51.509 --> 00:01:53.200
to be little y.
00:01:53.200 --> 00:01:55.690
But that's the same as
the random variable
00:01:55.690 --> 00:01:57.780
capital Y over 2.
00:01:57.780 --> 00:02:02.790
This is a random variable that
takes this value whenever
00:02:02.790 --> 00:02:07.400
capital Y happens to be
the same as little y.
00:02:07.400 --> 00:02:09.478
So the conditional
expectation--
00:02:09.478 --> 00:02:11.890
the abstract conditional
expectation is a random
00:02:11.890 --> 00:02:15.710
variable because its value is
determined by the random
00:02:15.710 --> 00:02:19.960
variable capital Y, and it is
this particular function of
00:02:19.960 --> 00:02:22.470
the random variable capital Y.
00:02:22.470 --> 00:02:25.820
And now we can proceed and
calculate the expected value
00:02:25.820 --> 00:02:30.150
of X using the law of iterated
expectations.
00:02:30.150 --> 00:02:35.060
The law of iterated expectations
takes this form.
00:02:35.060 --> 00:02:39.350
We have already calculated what
this random variable is.
00:02:39.350 --> 00:02:45.400
It is the random variable that's
equal to Y over 2.
00:02:45.400 --> 00:02:51.120
So this is the same as 1/2 the
expected value of Y. And since
00:02:51.120 --> 00:02:55.460
Y is uniform in the range from
0 to l, the expected value of
00:02:55.460 --> 00:03:00.880
Y is equal to l over 2,
which gives us an
00:03:00.880 --> 00:03:03.900
answer of l over 4.
00:03:03.900 --> 00:03:07.160
This is the same as the answer
that we got in the past where
00:03:07.160 --> 00:03:11.900
we actually found it using the
total expectation theorem.
00:03:11.900 --> 00:03:16.450
The calculations were exactly
the same as what went on here
00:03:16.450 --> 00:03:19.880
except that here we carry out
the calculation in a more
00:03:19.880 --> 00:03:21.310
abstract form.
00:03:21.310 --> 00:03:25.120
And what is important to
appreciate from this example
00:03:25.120 --> 00:03:29.470
is the distinction between
these two lines.
00:03:29.470 --> 00:03:33.970
This is an equality between
numbers, which is true for any
00:03:33.970 --> 00:03:36.140
specific little y.
00:03:36.140 --> 00:03:40.910
Whereas this is an equality
between random variables.
00:03:40.910 --> 00:03:44.650
This quantity is random and this
quantity is also random,
00:03:44.650 --> 00:03:47.730
meaning that their values are
not known until the experiment
00:03:47.730 --> 00:03:49.980
is carried out and the
specific value of
00:03:49.980 --> 00:03:51.329
capital Y is realized.