WEBVTT
Kind: captions
Language: en
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So let us see some simple example based on
the possible values of T and S.
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So, the first one is going to be a discrete
time or you can use a discrete parameter also.
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Discrete time, discrete state stochastic process
that means the possible values of S as well
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as the possible values of T has to be either
it has to be of countably finite or countably
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infinite elements in it.
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Let see the one simple example.
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Let us create a random variable X suffix n,
that is nothing but the number of customers
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in the Barber shop after nth customer departure
from the shop.
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So, here suffix n that will form a parameter
space therefore the T can be a possible value
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of n.
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That means whenever one customer leave the
system how many are in the system after he
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leaves.
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So the possible values of T will be the first
customers when he leaves out when he is not
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there he want to find out and so.
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Therefore, the possible values of T is going
to be 1, 2 or three therefore this is the
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number of making the number of customers in
the system.
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Whereas the possible values of Xn or possible
value of n that is going to be the –there
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is a possibility no customers in the system
when someone leaves.
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So, there is a possibility zero when someone
leaves only one customer in the system when
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it is going to be 1 or 2 and so on.
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Therefore, there is a possibility it could
be finite also.
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So the S can be countably finite or in this
case I have made the it is countably infinite.
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Therefore, the T as well as T is going to
be form of the discrete therefore the corresponding
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stochastic process Xn for possible values
of n is going to be 1, 2 and so on and this
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is going to be a discrete time, discrete state
to stochastic process.
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You please note that here the parameter space
T is not the time.
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The parameter space, forming the 1, 2, 3 these
are all the customers, the nth customers.
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Therefore, n can be 1, 2 and so on.
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Therefore, it usually the T is time whereas
sometimes it would be a distance or length
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or the number or whatever the other quantities.
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So here the typical situation in which the
parameter space is not considering the time.
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Therefore, this is going to be a random variable
because you never know how many customers
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are going to be in the system after the nth
customers leaves.
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Therefore, this is going to be a random variable.
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Obviously it is a real value function satisfying
all the properties of the definition and you
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can see the probability space for this and
from the probability space you have to create
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the random variable and therefore this random
variable is going to be the –this collection
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of random variable over the n that is going
to be the discrete time and discrete state.
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Therefore, this random variable here you can
create it with the help of a case one by making
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for fixed n, what is the random variable then
you make a collection of random variable.
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So, we can create this stochastic process
by using the case one or the approach one
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which the easier one.
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I can go for creating one more stochastic
process for this discrete time and the discrete
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state stochastic process that comes under
daily communication problems Xn is going to
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be number of packets waiting in the buffer
at nth time unit in the communication router.
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That means there is a communication router
in which the packets are coming for transmission.
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So after the transmission is over in the buffer
the packets are leave the router.
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So at any time you don not how many packets
are waiting in the buffer for the transmission.
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So there is a possibility no packets will
be there at some time point and there is a
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possibility there are many more packets may
be waiting for the transmission in the buffer.
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So, the possible values of S that is going
to be –there is a possibility no packets
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in the buffer or one or so on and similarly
the possible values of T that is also we are
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marking the nth time unit.
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Therefore, the time unit could view first
time unit or second time unit and so on.
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Therefore, here the S is going to be the discrete
as well as the T is going to be discrete therefore
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this collection of random variable X suffix
n for possible values of n that is also going
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to form a discrete time, discrete state stochastic
process.
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Because of the possible both the values are
going to be of discrete type discussing the
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simple stochastic process based on the parameters
ways and the state ways and we have seen the
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discrete time discrete state stochastic process.
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The first one now we are seeing the second
one.
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That is continuous time
discrete state stochastic process.
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That means the possible values of parameter
space is going to be a uncountably many values
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therefore we get the continuous time and the
possible values of the state based that is
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going to be a countably finite or countably
infinite therefore you get the discrete state.
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So, you will see the few simple example of
this type.
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The first example that is X of t that is going
to be the number of customers in the Barber
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shop at any time t that is difference.
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In the earlier, example we have seen the number
of customers in the barber shop for the nth
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customers departure now we are seeing the
number of customers in the barber shop at
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any time t.
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Therefore, we are looking at how many customers
at any time t in the barber shop.
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Therefore, the possible values T that is going
to be a collection of t such that the t is
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greater than or equal to zero.
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And the possible values of S that is going
to be still it is a number of customers therefore
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the possible values are 0, 1, 2 or it can
be when there is a possibility it can be countably
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finite also.
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So whether the state place is going to be
countably finite or countably infinite we
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classify as a discrete state.
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Therefore, this is a typical example of continuous
time discrete state stochastic process and
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the collection of random variable is going
to be X of t for all possibly values of t.
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So, this is going to form a real value the
stochastic process which for each t it is
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going to be a random variable.
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So, this is going to be a real value the stochastic
process of one dimensional type and the t
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is belonging to the T that is going to be
the time that is the default one and it is
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going to be a countably many therefore it
is going to be continuous parameter.
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So it is going to be call it as a continuous
parameter discrete state stochastic process
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also.