WEBVTT
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Language: en
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Welcome to the first lecture of the second
week of this course. Today we are going to
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discuss something called conditional expectation.
Conditional expectation is the first most
00:00:39.840 --> 00:00:44.880
important thing, that I need to discuss with
you. The remaining part, that I was discussing
00:00:44.880 --> 00:00:49.850
with you is something I was, expecting that
you had almost some idea about. But here we
00:00:49.850 --> 00:00:54.690
start with something very new usually not
done in standard courses of probability. Standard
00:00:54.690 --> 00:01:01.120
courses mean courses which engineers usually
take or economists take, etc.
00:01:01.120 --> 00:01:20.890
So, the exposition that I found most interesting
was from a book called Probability for Finance
00:01:20.890 --> 00:01:35.399
by Sean Dineen published by American Math
Society in 2005 under the series Graduate
00:01:35.399 --> 00:01:45.150
Texts in Mathematics. So, what does conditional
expectation mean? We will see, a very simple
00:01:45.150 --> 00:01:53.810
example. Suppose, I take this S is a random
variable or S is a random variable, which
00:01:53.810 --> 00:02:09.869
tells me, how many times head appear, when
I toss a coin three times. So, how many heads
00:02:09.869 --> 00:02:16.560
in three tosses of a fair coin, repeated of
course each, first toss, second toss, third
00:02:16.560 --> 00:02:35.050
toss, three repeated tosses of a fair coin.
00:02:35.050 --> 00:02:39.070
Of course, you know this is a binomial random
variable. We have already know, we know that,
00:02:39.070 --> 00:02:47.240
if N be the number of success and Np is the
expectation, given that p is the probability
00:02:47.240 --> 00:02:55.260
of success, sorry, how many heads, so just
make a little correction, how many heads in,
00:02:55.260 --> 00:03:01.500
three repeated tosses of a fair coin. So,
it tells you that how many successes you have.
00:03:01.500 --> 00:03:11.870
So, by success, I am meaning the appearance
of a head. So, I toss any of the three can
00:03:11.870 --> 00:03:15.500
come; head, head, head, head, tail, head and
all these things.
00:03:15.500 --> 00:03:20.760
So, you know that expectation of this because,
this is a binomial random variable, is nothing
00:03:20.760 --> 00:03:30.450
but n into p where probability of success
is half and n is 3 so 3 into half which is
00:03:30.450 --> 00:03:42.550
1.5. What happens if I now ask you this question,
find the expectation of S3? find the expected
00:03:42.550 --> 00:03:51.370
number of heads that you expect if you make
three repeated tosses of a coin, but with
00:03:51.370 --> 00:03:55.930
the knowledge now, that the first toss is
a head?
00:03:55.930 --> 00:04:07.360
So, now we are going to see how many tosses,
you have three repeated tosses, so how many
00:04:07.360 --> 00:04:11.730
heads would appear if you already know, how
many heads you are expecting rather, if you
00:04:11.730 --> 00:04:21.329
already know that the first toss is a head.
What is the answer. This is interesting.
00:04:21.329 --> 00:04:26.310
You will see now I know that the first toss
is a head. I really have to look into the
00:04:26.310 --> 00:04:38.830
second two positions, this is given. So, here
I can have head, head; head, tail; tail, head;
00:04:38.830 --> 00:04:48.740
and tail, tail. So, means, now I can have
only 1 head appearing remaining two are tail.
00:04:48.740 --> 00:04:56.650
I can have only two heads appearing there
is no tail at all so this one comes all here
00:04:56.650 --> 00:05:00.410
head comes and tail comes then tail comes
and head comes.
00:05:00.410 --> 00:05:11.730
So, if S3 is 1, if S3 can take value 1 it
can take value 2 it can take value 3. It can
00:05:11.730 --> 00:05:17.790
never take value 0 because, I know the first
one is 0 right. So, when I am talking about
00:05:17.790 --> 00:05:23.930
1 means these two tail has come. So here among
this four, now I do not bother about it, it
00:05:23.930 --> 00:05:29.270
is already a known fact so I just have to
look into this diagram. So, when tail comes
00:05:29.270 --> 00:05:35.020
both are tail so among these four choices
I have only one choice, so there my probability
00:05:35.020 --> 00:05:43.849
is one fourth. So, 1 into one fourth, right.
00:05:43.849 --> 00:05:56.139
Now if, I have the situation, where I have
2 heads which has come in three so among these
00:05:56.139 --> 00:06:01.569
two, one has been head; so, either head, tail;
tail, head. So, any one of the two should
00:06:01.569 --> 00:06:08.819
appear with probability of the 4, this will
appear two times, so the probability would
00:06:08.819 --> 00:06:20.180
be 1/2. You see the whole working changes.
Now 3 means I have got this. So, among all
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the 4 I have one chance. So, it is one fourth
plus one plus three fourth. So, it gives me
00:06:29.629 --> 00:06:39.069
one, so does this answer sorry 2, 1 plus 1
2, 1 plus 1 2, sorry this answer is 2.
00:06:39.069 --> 00:06:48.669
So, I leave you with the to show that, leave
you with this question, that if the first
00:06:48.669 --> 00:07:03.930
toss is tail then what is my expected occurrence
of heads? So, that would decrease. So, here
00:07:03.930 --> 00:07:08.620
my expectation increases. When the first one
is head, I am expecting that among the 2,
00:07:08.620 --> 00:07:13.659
3 at least 2 would be head. Here because the
first one is tail I am expecting at least
00:07:13.659 --> 00:07:20.150
among the 1 there will be 1 which will be
head. So, you see once more information is
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available my expectation changes. This is
called conditional expectation essentially.
00:07:28.220 --> 00:07:38.330
Now, how can I put this idea into a more rigorous
form. See, this idea is being developed on
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a scenario, where your omega is finite, your
sample space has finite elements. So, your
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Sigma-algebra or the set of all events, is
the power set of that finite set. So, let
00:07:51.890 --> 00:07:54.069
us look into this scenario.
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So, we have, instead of u I am writing F because,
that is what is usually written in finance
00:08:05.279 --> 00:08:21.569
books. So, let this be a probability space.
So, what is more interesting to know here,
00:08:21.569 --> 00:08:31.219
is that I take that this is finite. The cardinality,
when total number of elements of omega is
00:08:31.219 --> 00:08:49.080
finite, that is so my F is nothing but the
power set. So, F is the power set of omega.
00:08:49.080 --> 00:09:10.500
Now, assume that strictly greater than 0 for
all omega in the sample space. So, every sample
00:09:10.500 --> 00:09:14.010
point, every outcome of random experiment
the probability is strictly greater than 0,
00:09:14.010 --> 00:09:36.200
just like a head or tail scenario. Let
A be an event, such that probability of Aa
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is strictly bigger than and we can take it
less than 1 or strictly less than or equal
00:09:41.600 --> 00:09:47.280
to 1 does not matter much. So, this is what
I have assumed and under this assumption I
00:09:47.280 --> 00:09:50.870
have to define what is called a conditional
expectation.
00:09:50.870 --> 00:10:01.710
So, I am looking for the conditional expectation
of a random variable X, given that the event
00:10:01.710 --> 00:10:08.910
A has occurred. This is in some sense conditioning
using a Sigma-algebra because A belongs to
00:10:08.910 --> 00:10:14.050
a Sigma-algebra and soon we will come to that,
how we actually condition over a Sigma-algebra.
00:10:14.050 --> 00:10:30.570
This is nothing but the very definition, it
will be X of omega and probability of omega
00:10:30.570 --> 00:10:36.560
given the conditional probability of omega
given that the event A has occurred. This
00:10:36.560 --> 00:10:40.970
is exactly what will happen. This is exactly
the definition.
00:10:40.970 --> 00:10:53.140
So, what
00:10:53.140 --> 00:11:10.750
is probability omega intersection A. That
is
00:11:10.750 --> 00:11:26.310
probability of omega by probability of A,
if omega is in A and is 0 if, omega is not
00:11:26.310 --> 00:11:35.560
in A. This is quite simple to understand because,
you see what happens. When omega is in A,
00:11:35.560 --> 00:11:42.010
omega intersection A is omega, so you have
just P of omega by P of A, and when omega
00:11:42.010 --> 00:11:51.320
is not in A, omega intersection A is the null
event, empty, impossible event, so that time
00:11:51.320 --> 00:11:55.300
this will become 0 because, this will be the
null event.
00:11:55.300 --> 00:12:03.160
So now coming to this board again. I am rubbing
off what is in here. That is why sliding boards
00:12:03.160 --> 00:12:07.360
are so important and this place has space
for sliding boards actually. So, now I can
00:12:07.360 --> 00:12:17.130
write down the definition by putting in this
in this place. So, what I finally get is expectation
00:12:17.130 --> 00:12:38.890
X given that the event A has occurred is summation
omega element of A, X omega, P of omega divided
00:12:38.890 --> 00:12:48.470
by P of A, and this I can write as, 1 by P
of A and symbolically, you know that this
00:12:48.470 --> 00:13:02.540
sum is actually an integral if you were not
considering, the discrete distribution or
00:13:02.540 --> 00:13:06.010
discrete setup, then this would be an integral.
00:13:06.010 --> 00:13:13.310
So, essentially a sum, summation X omega,
P omega omega element of A but this we will
00:13:13.310 --> 00:13:18.400
always symbolically even if we have sum we
would symbolically write this as. So, this
00:13:18.400 --> 00:13:35.190
is just a symbolical writing for the expression
sorry is just a symbolical writing for the
00:13:35.190 --> 00:13:48.690
expression X Omega, P of omega, symbolical
writing.
00:13:48.690 --> 00:13:54.740
Of course, you can move to non-finite spaces,
which we will soon do, but this is just a
00:13:54.740 --> 00:14:00.770
useful trick to understand it. So, if you
want to write about, any other set, what about
00:14:00.770 --> 00:14:05.430
the compliment, if I know the compliment of
A has occurred. Then also, this is nothing
00:14:05.430 --> 00:14:17.240
but, the same you can actually prove that,
they are you can just replace A with.
00:14:17.240 --> 00:14:24.070
Now let us look at the Sigma-algebra which
the event A can generate, that is the smallest
00:14:24.070 --> 00:14:29.100
Sigma-algebra which contains the set A, it
should contain A, it should contain the empty
00:14:29.100 --> 00:14:37.720
set, it should contain the whole set, it should
contain its compliment. So, Sigma-algebra
00:14:37.720 --> 00:15:03.310
generated by A the empty set, the whole set
A and A compliment.
00:15:03.310 --> 00:15:16.610
So, if B, so we are just writing it B is element
of A so some sigma element B from this Sigma-algebra
00:15:16.610 --> 00:15:54.640
with element P B greater than 0, so we can
write. So now observe that, whenever B occurs,
00:15:54.640 --> 00:16:01.540
what does this means B occurs, some omega
belonged to B has occurred. When you have
00:16:01.540 --> 00:16:05.630
random experiment, what is the meaning of
B has occurred. When I say, I throw a dice,
00:16:05.630 --> 00:16:10.250
odd number has occurred. If 1 occurs odd comes
over the face, then the event that odd number
00:16:10.250 --> 00:16:12.360
has occurred has occurred.
00:16:12.360 --> 00:16:19.780
So, I can actually view this as a function.
So, for any omega that I take in B, I can
00:16:19.780 --> 00:16:31.780
define a function like this, rather a random
variable like this. So, for every omega in
00:16:31.780 --> 00:16:38.520
B, that is over the whole set B, whole event
B, this is the value. So, it remains constant
00:16:38.520 --> 00:16:43.500
over the whole event B, so if I look at it
like that, that I am looking at a function
00:16:43.500 --> 00:16:48.680
which is constant over various events, then
the conditional expectation itself can be
00:16:48.680 --> 00:16:51.640
viewed as a random variable.
00:16:51.640 --> 00:16:56.270
So, you see because it does not matter whatever,
whenever any omega occurs, B has occurred
00:16:56.270 --> 00:17:01.640
actually. So, for any omega in B this should
be the story.
00:17:01.640 --> 00:17:07.610
So, whatever once you know that, B has occurred
so for whatever omega you are taking this
00:17:07.610 --> 00:17:14.740
should be the story, this should be the answer.
So, this idea, that you can actually view,
00:17:14.740 --> 00:17:19.110
the conditional expectation itself as a random
variable is a very-very fundamental idea and
00:17:19.110 --> 00:17:24.980
is a very helpful idea. So, this is itself
a random variable r.v, where r.v. is just
00:17:24.980 --> 00:17:31.210
a short shorthand for random variables. So,
this will allow us to make shift from the
00:17:31.210 --> 00:17:34.740
standard probability space where we do not
bother about the finiteness of omegas and
00:17:34.740 --> 00:17:36.490
all those sort of things.
00:17:36.490 --> 00:17:54.910
So, first we will talk about a countable partition.
Now, what is the meaning of a countable partition
00:17:54.910 --> 00:18:08.550
of omega. So, consider a sequence of sets,
consider Gi, i is equal to 1 to infinity right,
00:18:08.550 --> 00:18:21.840
1 2 3 by G1 G2 Gn so consider Gi where G i
is a subset of omega for all i in N set of
00:18:21.840 --> 00:18:59.779
natural numbers. Then this Gi is called a
countable partition, if number 1, Gi intersection
00:18:59.779 --> 00:19:11.289
Gj is equal to phi for all i not equal to
j, and number 2, union of Gi, i is equal to
00:19:11.289 --> 00:19:17.139
1 to infinity, should give me back omega the
whole sample space.
00:19:17.139 --> 00:19:23.620
So, this is called a countable partition and
what we would now require, is a Sigma-algebra
00:19:23.620 --> 00:19:28.120
generated by that countable partition that
you take elements of that countable partition,
00:19:28.120 --> 00:19:34.200
when I basically, take, do not take intersections,
because intersections would ultimately generate
00:19:34.200 --> 00:19:39.029
either the, if you take intersection of the
same set it will generate the same set itself
00:19:39.029 --> 00:19:41.679
or it will generate the empty set.
00:19:41.679 --> 00:19:48.999
So basically, if you want to take any non-empty
set of a Sigma-algebra generated by the set,
00:19:48.999 --> 00:19:54.759
you just have to construct unions of this
set using some subset of n could be finite
00:19:54.759 --> 00:19:57.379
could be infinite whatever right.
00:19:57.379 --> 00:20:06.700
So, what we will be interested is, in a Sigma-algebra
generated by a countable partition, which
00:20:06.700 --> 00:20:21.860
will be a sub Sigma-algebra of F, naturally
because, we are taking a subset, when taking
00:20:21.860 --> 00:20:26.100
a subset of A and generating a Sigma-algebra.
00:20:26.100 --> 00:20:44.759
So, consider a Sigma-algebra G, subset of
f and G, G being generated by a countable
00:20:44.759 --> 00:21:05.730
partition. Then, we define it like this, and
the conditional expectation of the random
00:21:05.730 --> 00:21:16.499
variable X, conditioned on the Sigma-algebra
G, is defined as like this, at any omega right,
00:21:16.499 --> 00:21:35.710
say omega n or whatever, any omega is P Gn
integral Gn X dP because this is a countable
00:21:35.710 --> 00:21:39.269
because this is a generated by a countable
partition say G i.
00:21:39.269 --> 00:21:49.039
So, given any omega it must lie in one of
the Gi s because omega is the element of omega
00:21:49.039 --> 00:21:54.619
and this is a countable partition of omega
so it must lie in one of the Gi s. So, suppose
00:21:54.619 --> 00:22:01.559
take any omega then this will happen. This
is the way I define how you compute this random
00:22:01.559 --> 00:22:15.470
variable at the point omega if omega is element
of G n, good.
00:22:15.470 --> 00:22:22.279
I will now, been a random variable, it is
important to know that because I am claiming
00:22:22.279 --> 00:22:33.030
it to be a random variable, you should be
able to prove that, I leave this as an exercise
00:22:33.030 --> 00:22:40.480
to you, in your assignments, so of course
and this is true. You are going to prove this
00:22:40.480 --> 00:22:48.740
fact that this is measurable.
An interesting thing is that if x is integrable
00:22:48.740 --> 00:22:56.870
random variable, that is it has a finite expectation
then this random variable also has a finite
00:22:56.870 --> 00:23:08.389
expectation that is the, conditional expectation,
conditioned on a sub Sigma-algebra of F which
00:23:08.389 --> 00:23:17.809
is generated by a countable partition, has
also got to be integrable. That can be proved,
00:23:17.809 --> 00:23:23.649
but I would not rather prove this fact now.
00:23:23.649 --> 00:23:33.450
So, this would be a part of the exercise.
If X is integrable, I am expecting that, you
00:23:33.450 --> 00:23:38.710
know what is integrability though we have
given some basic definitions. So, if X is
00:23:38.710 --> 00:23:50.860
integrable then, integrable means it has a
finite expectation because expectation is
00:23:50.860 --> 00:24:19.999
expressed in terms of an integral, that is
integral X dP this is finite, then E is integrable.
00:24:19.999 --> 00:24:29.340
That is this random variable also has a finite
expectation. This is important.
00:24:29.340 --> 00:24:36.989
Now, once we know this, we are now going to
state a very important property, of this random
00:24:36.989 --> 00:24:44.639
variable. This property essentially characterizes
and this idea actually allows us to move beyond
00:24:44.639 --> 00:24:52.600
Sigma-algebras, which are Sigma-algebras of
this type, that is you will now, once you
00:24:52.600 --> 00:24:58.020
know this you can, once we know this result
what we are going to state, using this idea,
00:24:58.020 --> 00:25:03.309
we can give a very general definition of a
random, conditional expectation as a random
00:25:03.309 --> 00:25:12.640
variable that is we can define the conditional
expectation, conditioned not just by a Sigma-algebra
00:25:12.640 --> 00:25:17.299
generated by a countable partition, but by
any Sigma-algebra which is a sub Sigma-algebra
00:25:17.299 --> 00:25:18.620
of F.
00:25:18.620 --> 00:25:28.559
So, you are writing this important property,
give a very small proof of this and then we
00:25:28.559 --> 00:25:36.029
will wind up today’s discussion. So, this
section is the first part of conditional expectation
00:25:36.029 --> 00:25:41.970
and tomorrow we are going to discuss the general
case and the properties of conditional expectation
00:25:41.970 --> 00:25:43.389
that is it.
00:25:43.389 --> 00:26:05.110
So, what does this proposition say? So again,
G is a sub Sigma-algebra of F, generated by
00:26:05.110 --> 00:26:43.120
a countable partition, Gi 1 to infinity. Assume,
of course this definition when you are writing,
00:26:43.120 --> 00:26:52.100
you are assuming that P Gn is strictly bigger
than 0 for all N because without that you
00:26:52.100 --> 00:26:53.860
cannot write this.
00:26:53.860 --> 00:27:04.009
Assume that P Gn that is probability of each
of these pieces of the partition is greater
00:27:04.009 --> 00:27:25.940
than equal to 0 for all N. Then E X by G this
conditional expectation is the unique
00:27:25.940 --> 00:27:33.529
G measurable, of course when I am talking
about measurability here, I am essentially
00:27:33.529 --> 00:27:50.600
talking about G measurability, is the unique
G measurable random variable
00:27:50.600 --> 00:28:14.710
such that, random variable on the probability
space of course, such that for all A in G.
00:28:14.710 --> 00:28:24.450
So, on G, integrating over a set of G, subset
of G, integrating over any element in G, any
00:28:24.450 --> 00:28:30.419
set which belongs to G, over X is same as,
integrating over the conditional expectation.
00:28:30.419 --> 00:28:38.510
So, it is essentially it says that, over G
conditional expectation and X are almost the
00:28:38.510 --> 00:28:45.830
same thing. Over such as Sigma-algebra, X
and conditional expectations, they behave
00:28:45.830 --> 00:28:52.779
in a almost similar fashion. The integrals
are same and this idea is actually used to
00:28:52.779 --> 00:28:56.759
extend, to a higher basically, when you go
to a case when you do not take this countable
00:28:56.759 --> 00:29:00.590
partition you take this as a definition of
condition expectation.
00:29:00.590 --> 00:29:08.200
You see this is how mathematics develops because,
you do it for some simple case and then you
00:29:08.200 --> 00:29:12.669
know that, it is not so easy to talk about
the general case, so you pull it up by taking
00:29:12.669 --> 00:29:17.270
the, what you know for the simple case as
a general definition and that definition will
00:29:17.270 --> 00:29:21.980
always work if the case is simple. So, we
will come to that later on. It has lot of
00:29:21.980 --> 00:29:26.720
links with certain things called Radon-Nikodym
theorem. So, let us give a proof of why this
00:29:26.720 --> 00:29:27.720
is happening.
00:29:27.720 --> 00:29:36.200
I am not going to prove, the case for uniqueness
because, that is very simple. So, to prove
00:29:36.200 --> 00:29:41.980
the uniqueness, we really have to use take
any other Y, which for this the same thing
00:29:41.980 --> 00:29:49.369
happens, which will tell me that integral
of this variable minus Y dP would be 0. So,
00:29:49.369 --> 00:29:59.430
which means that if this function is 0, what
does that mean, function is 0 almost, that
00:29:59.430 --> 00:30:05.360
is exactly. So, that is the how you receive
your uniqueness. So, we will just do the existence
00:30:05.360 --> 00:30:12.710
of this that okay if I define the random variable
in this way the way which we have defined
00:30:12.710 --> 00:30:22.710
in this particular case then this must be
equal to this. So, proof and with that we
00:30:22.710 --> 00:30:28.100
will end today’s discussion.
00:30:28.100 --> 00:30:36.429
So, I am just taking with this first with
the Gn, let us see why we are doing so. First,
00:30:36.429 --> 00:30:48.259
you have to understand why I am just taking
first with Gn because, if A is any non-empty
00:30:48.259 --> 00:31:02.739
set in G, then A is always written as, union
M subset of N Gn where M is a subset of N,
00:31:02.739 --> 00:31:06.009
could be finite could be infinite whatever.
00:31:06.009 --> 00:31:16.580
So, take some Gn from there and you are writing
this Gn, M is capital M is a set. It contains
00:31:16.580 --> 00:31:22.460
the indexes which is subset could be just
1 2, 3, 4, 5 so any subset of M. So, every
00:31:22.460 --> 00:31:27.019
A can be expressed like this right because
the intersection you could take intersection
00:31:27.019 --> 00:31:30.789
of the empty right that is why you cannot.
So, every non-empty set has to be expressed
00:31:30.789 --> 00:31:32.389
like this okay.
00:31:32.389 --> 00:31:43.441
Let us see what happens on this Gn. Now you
observe, that on this Gn for every part, if
00:31:43.441 --> 00:31:50.409
you look at the definition this is a fixed
quantity. You take any omega n element of
00:31:50.409 --> 00:31:54.299
Gn, whatever omega you take it will give you
the same value. It is a constant on that Gn
00:31:54.299 --> 00:32:12.700
part. So, I can take to be some omega integral
Gn dP, that omega is element of G n.
00:32:12.700 --> 00:32:18.790
Now what is this. This is nothing but probability
of Gn but I can again use this, I will write
00:32:18.790 --> 00:32:31.840
down the definition here, probability of Gn
integral Gn X dP probability of Gn because
00:32:31.840 --> 00:32:37.190
this is nothing but probability of Gn integral
d P Gn is probability of Gn. So, what I will
00:32:37.190 --> 00:32:42.570
get here is integral Gn X dP.
00:32:42.570 --> 00:33:00.070
Now if I am talking about integral A, so I
can write this as integral union of Gn M subset
00:33:00.070 --> 00:33:14.629
of N because, these are all disjoint, I can
actually sum them up sum each of the integrals.
00:33:14.629 --> 00:33:32.229
So, it is summation M subset of N and summation
of all the indexes of N integral Gn and this
00:33:32.229 --> 00:33:36.600
summation, what is this, for each individual
piece this is nothing but this. So, this is
00:33:36.600 --> 00:33:49.070
nothing but summation M subset of N integral
Gn X dP, which is nothing but, integral union
00:33:49.070 --> 00:33:58.139
of Gn, union of Gn is nothing but A which
is integral A X dP and that is the answer
00:33:58.139 --> 00:33:59.450
that is the proof.
00:33:59.450 --> 00:34:05.210
So, with this proof we end our discussion
here today.
00:34:05.210 --> 00:34:09.460
Thank you very much.