WEBVTT
Kind: captions
Language: en
00:00:07.759 --> 00:00:29.640
This
00:00:29.640 --> 00:00:36.780
course needs prerequisite of a probability
as a full one semester course.
00:00:36.780 --> 00:00:41.370
So most of the universities they have a course,
probability theory along with the stochastic
00:00:41.370 --> 00:00:45.969
processes or random process or probability
and statistics.
00:00:45.969 --> 00:00:52.440
So whatever the courses we have, at least
some 30 lectures of probability theory is
00:00:52.440 --> 00:00:58.019
needed for this stochastic process course
as a prerequisite.
00:00:58.019 --> 00:01:05.059
Other than probability course we need basic
course in calculus and some mathematical background
00:01:05.059 --> 00:01:10.610
over the combinatorial problems and also the
matrix algebra.
00:01:10.610 --> 00:01:15.790
So these courses would have been covered in
the max one or mathematics two courses.
00:01:15.790 --> 00:01:35.780
So that is enough to understand the stochastic
process course.
00:01:35.780 --> 00:01:40.100
So before we move into the stochastic process,
I am going to give what is the motivation
00:01:40.100 --> 00:01:42.240
behind the stochastic process.
00:01:42.240 --> 00:01:51.930
When we see the last few decades problems,
more of the probability models are not deterministic
00:01:51.930 --> 00:01:57.860
that means you need more probability theory
to understand the stochastic to understand
00:01:57.860 --> 00:02:03.150
the system then only you can study the dynamics
of the model.
00:02:03.150 --> 00:02:08.200
If you see the, if you want to study the dynamics
of the system, then you need more probably
00:02:08.200 --> 00:02:09.280
theory.
00:02:09.280 --> 00:02:17.230
So the simple probability theory may not be
enough to study the more study on the realistic
00:02:17.230 --> 00:02:18.560
system.
00:02:18.560 --> 00:02:24.150
The way this realistic system behaves in a
very dynamical way, it is not easy to capture
00:02:24.150 --> 00:02:28.689
everything through the probabilistic or usual
probability models.
00:02:28.689 --> 00:02:35.629
That means you need more than the probability
theory to understand the system or to study
00:02:35.629 --> 00:02:38.230
the system in a well behaved way.
00:02:38.230 --> 00:02:42.499
For that one of the important thing is stochastic
process.
00:02:42.499 --> 00:02:48.459
It deals about the collection of a random
variable, so that you can study the dynamics
00:02:48.459 --> 00:02:50.950
of the system in a better way.
00:02:50.950 --> 00:02:55.989
Even though I am giving very light way of
saying the collection of random variable first
00:02:55.989 --> 00:03:01.900
we should know how the random variable can
be defined, so that you can study the collection
00:03:01.900 --> 00:03:04.329
of random variable in a better way.
00:03:04.329 --> 00:03:11.209
So for that we are going to spend few examples
through that how the more realistic models
00:03:11.209 --> 00:03:16.159
need more probability theory other than the
usual probability theory.
00:03:16.159 --> 00:03:21.480
So that the stochastic process definition
and those things I am going to cover at a
00:03:21.480 --> 00:03:22.480
later part.
00:03:22.480 --> 00:03:29.779
First let us see the first example, that comes
in the finance situation.
00:03:29.779 --> 00:03:38.280
This is the actual data which captured over
the period of time from August 01, 2009 to
00:03:38.280 --> 00:03:47.639
December 31, 2009 of what is the current price
of one US dollar in Indian rupees.
00:03:47.639 --> 00:03:55.650
So if you see the graph you can make out August
1st, 2009, the price of one US dollar was
00:03:55.650 --> 00:04:02.569
Rs. 47.57 or Rs. 47.58.
00:04:02.569 --> 00:04:09.689
And if you see the dynamics over the years,
over the days from August 01st, 2009 to till
00:04:09.689 --> 00:04:17.310
December 31st, 2009 it keeps on changing and
it takes some values higher and after that
00:04:17.310 --> 00:04:20.880
it goes down and it fluctuates and so on.
00:04:20.880 --> 00:04:29.530
So this is the actual data which we have captured.
00:04:29.530 --> 00:04:35.850
And from that our interest will be what could
be the US dollar price after sometime.
00:04:35.850 --> 00:04:40.980
If I know till today what is the price, my
interest would be what would be the price
00:04:40.980 --> 00:04:45.370
after 1 or 2 days or after one month or after
6 months.
00:04:45.370 --> 00:04:51.620
That means, I should know how the dynamics
keep moving over the days and what is the
00:04:51.620 --> 00:04:59.230
hidden probabilistic distribution is capturing
over the time, so that I can identify what
00:04:59.230 --> 00:05:01.020
is the distribution behind that.
00:05:01.020 --> 00:05:07.260
Therefore, I can study the future prediction;
I can study the dynamics of this particular
00:05:07.260 --> 00:05:09.380
model in a much better way.
00:05:09.380 --> 00:05:14.970
That means I need what is the background or
what is the hidden distribution playing or
00:05:14.970 --> 00:05:22.060
hidden distribution which causes the dynamics
of the system.
00:05:22.060 --> 00:05:27.530
After identifying what is the distribution,
my interest could be, what could be the some
00:05:27.530 --> 00:05:29.240
other moment over the time.
00:05:29.240 --> 00:05:35.420
That means what would be the average value
or what could be the second order moment if
00:05:35.420 --> 00:05:42.630
it exists and so on, that can be obtained
if I know the actual distribution in the underlying
00:05:42.630 --> 00:05:44.160
model.
00:05:44.160 --> 00:05:51.400
If I see the second example, I am just changing
into another model in which there are two
00:05:51.400 --> 00:05:53.960
people playing a game.
00:05:53.960 --> 00:05:56.770
The person A and person B.
00:05:56.770 --> 00:06:04.210
Whenever the person A wins he gets rupees
one.
00:06:04.210 --> 00:06:11.010
suppose the person B wins then he will get
the one rupee and at the same time person
00:06:11.010 --> 00:06:18.520
A loses one rupee in the same way and the
play is keep going.
00:06:18.520 --> 00:06:33.250
Suppose you make the random variable as xn
is the amount of the person A has at the end
00:06:33.250 --> 00:06:41.140
of nth game.
00:06:41.140 --> 00:06:48.460
If you make out the random variable xn for
the person A has the amount at the end of
00:06:48.460 --> 00:06:49.460
the nth game.
00:06:49.460 --> 00:06:57.710
The way the game going on, the value of the
xn keep changing and if you make out another
00:06:57.710 --> 00:07:05.940
random variable, Sn is the sum of xi where
i is running from one to n.
00:07:05.940 --> 00:07:23.930
This gives what is the total amount of the
person A has.
00:07:23.930 --> 00:07:31.770
The diagram in which the Sn gives, what is
the way the dynamics goes and over the n.
00:07:31.770 --> 00:07:36.230
And if you see the diagram you can make out
the whole dynamics goes.
00:07:36.230 --> 00:07:41.740
How the game is going on in the first few
games, accordingly it changes the positive
00:07:41.740 --> 00:07:44.210
side or it goes to the negative side.
00:07:44.210 --> 00:07:50.170
And if the n goes large then the dynamics
of the Sn over the n will keep changing over
00:07:50.170 --> 00:07:57.130
the time and you will get the realization
of the Sn over the time.
00:07:57.130 --> 00:08:06.970
And here I have given 3 different realizations
and this diagram is taken out from the book
00:08:06.970 --> 00:08:08.500
by U.N. Bhatt.
00:08:08.500 --> 00:08:11.870
The title of the book is elements of applied
stochastic process.
00:08:11.870 --> 00:08:16.370
So this is one of the motivations behind stochastic
process.
00:08:16.370 --> 00:08:22.580
And from these our interest will be after
the, what is the distribution of Sn at any
00:08:22.580 --> 00:08:26.889
n and also as n tends to infinity what could
be the distribution of Sn.
00:08:26.889 --> 00:08:32.690
That means you need the distribution of the
random variable and also you need what could
00:08:32.690 --> 00:08:37.329
be the distribution as n tends to infinity
or the limiting distribution of Sn.
00:08:37.329 --> 00:08:43.180
If you know the distribution, then you can
get all other moments for different n as well
00:08:43.180 --> 00:08:49.240
as the asymptotic behavior of the random variable
Sn.
00:08:49.240 --> 00:08:54.820
Next I will move into another example in which
it is the queuing situation.
00:08:54.820 --> 00:09:00.480
The queuing situation, here I have taken it
as, taken a simple example, that is a barber
00:09:00.480 --> 00:09:15.500
shop example, in which there is only one barber
shop person and who does the service for the
00:09:15.500 --> 00:09:18.930
people whoever entering into the barber shop.
00:09:18.930 --> 00:09:25.180
And there are only a limiting capacity in
which there is a maximum 10 people can stay
00:09:25.180 --> 00:09:28.440
in their barber shop and one person will be
under service.
00:09:28.440 --> 00:09:35.850
Once the service is over and the system will
be, the customer can leave the system.
00:09:35.850 --> 00:09:46.120
At any time, maximum 10 people can be in the
barber shop and only one person is doing the
00:09:46.120 --> 00:09:49.810
service for the customer who enter into the
barber shop.
00:09:49.810 --> 00:10:01.000
Suppose you take the random variable as x(t)
is the number of customers in the barber shop
00:10:01.000 --> 00:10:14.300
or in the system at time t the way the dynamics
goes; the possible values of x(t) will be
00:10:14.300 --> 00:10:17.490
starting from 0 to n.
00:10:17.490 --> 00:10:23.810
To study this system, you need what is the
way the people or the customers are entering
00:10:23.810 --> 00:10:31.440
into the system and what is the way the service
is going on for the customers and what is
00:10:31.440 --> 00:10:37.480
the discipline in which the customers are
getting served also.
00:10:37.480 --> 00:10:42.370
Our interest will be, suppose we have the
capacity of 10, what could be the waiting
00:10:42.370 --> 00:10:45.430
time, whenever the customers are entering
into the system.
00:10:45.430 --> 00:10:57.260
My interest will be one is how to reduce the
waiting time on average, this is the customers
00:10:57.260 --> 00:10:58.260
point of view.
00:10:58.260 --> 00:11:06.890
As the barber shop point of view how much
I can get more revenue, that means how I can
00:11:06.890 --> 00:11:12.990
increase the capacity of the systems so that
I can make more profit over the time.
00:11:12.990 --> 00:11:23.740
That means if I know the dynamics of x(t)
over t where t is varying from 0 to infinity,
00:11:23.740 --> 00:11:28.970
I can understand the system over the time
as well as I can whatever the probabilistic
00:11:28.970 --> 00:11:34.270
measures or whatever the other measures average
number of customers or average waiting time
00:11:34.270 --> 00:11:35.270
and so on.
00:11:35.270 --> 00:11:38.310
I can find out using this type of random variable.
00:11:38.310 --> 00:11:45.290
So later we are going to say, this is going
to be one of the stochastic process for this
00:11:45.290 --> 00:11:46.830
example.
00:11:46.830 --> 00:11:53.110
Next I am going to consider the fourth example
as the telecommunication system.
00:11:53.110 --> 00:11:59.700
Suppose you think of a system in which you
have n trunks are there.
00:11:59.700 --> 00:12:06.310
Trunks are nothing but it is a maximum number
of calls will be allowed at anytime.
00:12:06.310 --> 00:12:12.980
Whenever a call entering into the system and
you have given one trunk to the call and at
00:12:12.980 --> 00:12:16.080
the end of the call is over the trunk will
be back.
00:12:16.080 --> 00:12:22.620
So you have a telecommunication system in
which n trunks are available at any time not
00:12:22.620 --> 00:12:26.000
at any time just n trunks available.
00:12:26.000 --> 00:12:41.270
Suppose I make a random variable x(t) as the
number of calls on going, at time t, see here
00:12:41.270 --> 00:12:49.510
also the dynamics of x(t) is going to be keep
changing from 0 to n over the time and my
00:12:49.510 --> 00:12:57.380
interest will be how I can do the service
such a way that more calls will be entertained
00:12:57.380 --> 00:13:08.770
as well as how I can find out the optimal
n such a way that what is the optimal number
00:13:08.770 --> 00:13:22.170
of trunks such that I can minimize the waiting
time or I can maximize the revenue.
00:13:22.170 --> 00:13:28.280
So this is also one of the problems which
we come across in the daily life and so on.
00:13:28.280 --> 00:13:33.980
So my interest is to introduce the stochastic
process so that I can study this type of system
00:13:33.980 --> 00:13:34.750
in a better way.