WEBVTT
Kind: captions
Language: en
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The next example.
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The example two.
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In the example two, when we make X of t that
is going to be number of customers eating
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food in a restaurant
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at any time t.
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Therefore, you are observing the system you
are observing the restaurant.
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How many customers are taking their food?
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Therefore, the possible values of parameter
T is going to be t greater than or equal to
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zero.
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And the possible values of S still it is a
count therefore the possible values are accountably
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finite or countably infinite.
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Therefore, this collection of random variable
over the t that is going to be a continuous
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time or continuous parameter discrete state
stochastic process.
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This is a very typical example so it could
be countably finite or countably infinite
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also.
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Now let us see the third type that is discrete
time continuous state stochastic process
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that means we need to have the T value has
to be countably finite or countably infinite.
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Whereas the possible values of state space
has to be a countably many of that type.
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So, let us create example for that.
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The X suffix n is nothing but it is a random
variable that denotes the content of a dam
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or water reservoir observed at nth time unit.
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So, here the time unite could be every one
hour or that could be because you are seeing
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what is the content of dam or water reservoir.
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It could be everyday fixed time of everyday
or it could be fixed time of weekly once so
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that is going to be the time.
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So that is going to be the time period.
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So at the end of each nth time unit we are
observing what is the content of dam.
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So that is nothing but it is real quantity.
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Therefore, T is going to be you are observing
only at that time unit.
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So, either it would be one or daily once or
weekly once or so on.
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So, therefore I can make a one to one correspondence
with the countably finite or countably infinite
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numbers so that they will form a parameter
space and the S this is going to be possible
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value of Xn for all possible values of n.
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So, this is water content of dam that is going
to be the real quantity that is going to be
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for some x where x is always greater then
equal to zero.
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So, that means parameter space is going to
be a discrete whereas the state space is going
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to be a continuous therefore these two stochastic
process X suffix n for possible values of
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n is going to be 1, 2, 3, 4 and so on and
this is going to form discrete time continuous
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state stochastic process.
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Let me give one more example for the same
type that is example two
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That is nothing but example two Xn is nothing
but the amount of 1 US dollar in rupees at
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n th time unit in a day.
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That means I am just observing what is the
value of 1 US dollar in rupees in a day for
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the n th time unit.
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It could be every five minutes or it would
be every minute or it could be every hour
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of any particular day and that is going to
form a random variable and that collection
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is going to form a stochastic process.
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In, this the possible values of X is going
to be since it is the amount of 1 US dollar
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in rupees it could be some fraction also.
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Therefore, you do not want to take it as the
integer number.
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It could be real numbers therefore it is going
to a possible values of x greater than or
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equal to zero and the T that is going to the
time unit either it is every minute of every
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once in five minute or once in ten minutes
or every one hour or so on.
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So, this is going to form a countably finite
or countably infinite one and this stochastic
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process will form a discrete time and continuous
state stochastic processes.
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Let me go for the fourth type
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That is the fourth classification of the Stochastic
Process that is continuous time, continuous
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state stochastic process that means the possible
values of the parameter is going to be uncountably
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many therefore you get the continuous time
or continuous parameter and the possible values
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of the state base that is going to be uncountably
many therefore you get the continuous state
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stochastic process.
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The examples or the first one X t is going
to be temperature
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of a particular city at any time t.
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So whenever I use anytime t you can take any
value therefore the possible value of S is
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going to be the temperature so you can think
of temperature suppose a particular city lies
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between minus 50 to 60 degree Celsius So this
quantity is going to be the Celsius of minus
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50 to positive 60 and the parameter space
t is going to be observed over the time.
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Therefore, this time is going to be greater
than or equal to zero.
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Therefore, the parameter space is continuous
one and the state space is continuous one
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therefore this collection or random variable
form continuous time continuous state stochastic
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process.
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Let me give an another example of fourth type
, that is example 2.
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X(t) is the content of dam observed at any
time t.
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so the content of dam reservoir that is going
to be the real quantity therefore S is going
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to be a collection of x such that x is going
to be greater than equal to zero.
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And you are observing over the time therefore
that is also collection of t such that t is
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going to be greater than or equal to zero.
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Therefore, this will form a stochastic process
in which it will be under that classification
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of continuous time, continuous space stochastic
process and this can be created with the help
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of the first approach that means for fixed
t find out what is the random variable and
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you collect the random variable over the all
possible values of t.
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Therefore, this is going to be of the continuous
time and continuous state stochastic process.
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So in this lecture what we have seen what
is the meaning of
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stochastic process or how to create the stochastic
process so that is nothing but it is going
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to be a collection of random variables.
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So, we have defined stochastic process as
well how to create then later we have given
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what is parameter space and what is state
space and we have given the classification
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of stochastic process based on the parameter
space and state space and also some of the
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real world problems from that we can create
stochastic process.
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And that stochastic process and there are
many more stochastic process can be created
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with the help of the definition and so on
so that will be discussed in the lecture two.
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And these are all reference
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books we have used it for preparing this lecture
one material.
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And I will be continuing the lecture two with
the some more stochastic process which is
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very useful in later stages.
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Thank you.