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Hello, welcome to another module in this massive
open online course on the principles of probability
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and random variables for wireless communications.
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So in the previous model, we were looking
at the concept of independence and independent
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events.
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We had defined the concept, we defined mathematically
what does it mean to say that the 2 events
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A and B are independent.
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Today let us continue to look at the other
examples of independence.
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For instance, we had looked in the previous
module.
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So we were talking about independent events.
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We are currently focusing on independent events
and we had looked at this particular example
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where, we had looked at our sample space consisting
of 4 PAM modulation that is minus 3 alpha,
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minus alpha, alpha, and 3 alpha.
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We had said that this is our 4 PAM or M PAM,
M-ary PAM, pulse amplitude modulation constellation.
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This is our 4 PAM constellation and we had
considered the probabilities.
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1 by 8, 1 over 4 and half and we had also
looked at 2 events.
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Corresponding to the 1st event is A which
is the symbol minus 3 alpha or alpha.
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That the symbol is either minus 3 alpha or
alpha and B which is basically that the symbol
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is alpha.
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And we had realized that, we had rigorously
proved that the event, this event A where
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the symbol is either minus 3 alpha or alpha
and this event B where the symbol is alpha
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, these 2 events, A and B are not independent.
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So we had established in the last module that
these 2, A, B are not independent.
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These are not independent.
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So we had realized that these 2 events, A
and B are not independent.
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Then one would like to ask, what is the relevance
of independence in communication?
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What is the relevance of independence in digital
communication or wireless communication?
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What is the relevance of this concept of independence
and independent events in a wireless communication
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system?
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The relevance is that we had looked previously
at a single symbol.
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Remember, when we looked at the previous examples,
we had talked in the context of a single symbol
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belonging to the set A that the event A and
the event B in the context of a single symbol.
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However now, we will start looking at the
concept of different symbols generated.
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Because we have a wireless communication system,
so the source is generating a stream of output
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communication symbols.
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Alright ! So when we look at these different
symbols which are generated at successive
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instance of time, these different symbols
in the digital communication system or the
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wireless communication system are frequently
independent.
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So what you would like to say is that the
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symbols generated at different time instants
are independent.
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All right?
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So the symbols that are generated at different
time instants in digital communication or
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a wireless communication system are frequently
independent.
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I am not saying that this is always the case
but frequently, the successive symbols, the
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information symbols, the successive digital
modulation symbols that are generated by the
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communication system, these are frequently
independent.
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So let us see what this means.
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For example, let X1 denote the 1st symbol
that is generated at time interval 1.
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Let X2 be the 2nd symbol.
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Then X1, X2 are independent.
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Right?
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We are saying that X1 is the symbol that is
generated at time instant 1 and X2 is the
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symbol generated at time interval 2.
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Both of these can be drawn from the same digital
modulation constellation.
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For instance, in this example, let us say,
X1 and X2 are both symbols that are drawn
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from the 4 PAM constellation.
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That is X1 belongs to the set, minus 3 alpha,
minus alpha, alpha, and 3 alpha.
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X2 also belong to same set that is minus 3
alpha, minus alpha, alpha, and 3 alpha.
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And we are saying that these two 4 PAM symbols,
X1 and X2 are drawn independently from this
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sample space corresponding to the M-ary PAM
or the 4 PAM, the 4 pulse amplitude modulation
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constellation.
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So we are saying.
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This X1 and X2 are symbols generated independently
from the 4 PAM constellation.
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Alright ? In fact, to extend this analogy
further, we can consider a source S and frequently
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this is the case.
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That is we can consider a source.
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This is generating the symbols X1, X2, X3
and so on.
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So the source is generating various symbols.
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Each Xi, that is the ith symbol, symbol generated
at the ith time instant, each Xi is generated
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we say IID with stands for independent
identically distributed from capital asset
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is our 4 PAM constellation that is minus 3
alpha, minus alpha, alpha, and 3 alpha.
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So what we are saying is, not only X1 and
X2, all of these symbols, in communication
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system, it is continuously generating a stream
of information symbols.
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Right?
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Continuously generating a stream of modulated
symbols.
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We are saying these symbols, X1 , X2, X3É.
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Xi, all of these symbols are independent.
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And not only that, they are identically distributed.
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Each is generated independently and has the
identical distribution as the other in the
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sense that they belong to the 4 PAM constellation
that is the set S with the symbols minus 3
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alpha, minus alpha, alpha, and 3 alpha.
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And they have the same probability distribution.
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So each of these symbols are generated and
these are independent and identically distributed
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according to the 4 PAM, from the 4 PAM constellation.
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So these symbols are generated independent
identically.
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Let us take a simple example to understand
this.
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Let us look at a simple example to understand
this thing.
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Let us ask the question, if X1, X2 are independent,
what
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is the probability that X1 is equal to alpha
and X2 is equal to 3 alpha.
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So we are asking the following question, we
are looking at a simple example, we are saying,
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X1 and X2 are 2 symbols that are generated
by this 4 PAM source.
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So these are the 2 PAM that are generated
by this 4 PAM source and we are asking this,
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we are given that X1 and X2 are generated
independently.
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And we are asking the question that what is
the probability that X1 is equal to alpha
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and X2 is equal to 3 alpha?
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So these are generated independent and identically
from our 4 PAM source.
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So we are asking, what is the probability,
X1 is equal to alpha, X2 is equal to 3 alpha?
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That is X1, X2 is equal to alpha, 3 alpha.
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And this answer is as follows.
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The probability that X1 equals Alpha, X2 is
equal to 3 alpha.
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And since we said that these 2 symbols are
generated independently, now we can use the
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property of independent events.
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That is the joint probability or the probability
of the intersection is the product of the
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individual probabilities.
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Therefore, this is equal to the probability
X1 equals Alpha times the probability X2 equals
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3 alpha and as we have given the values initially
the probability that X1 is equal to alpha,
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this is equal to 1 by 4.
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The probability X2 is equal to 3 alpha.
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This is half.
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So this is basically equal to 1 over 8.
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So this is a simple problem where we are saying
what is the probability that we have 2 symbols
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X1 and X2 and these are generated independently
and we are saying, what the probability that
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X1 is equal to alpha and X2 is equal to 3
alpha.
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We are saying, the probability of this joint
event is equal to the product of the probabilities
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since these 2 symbols are independent which
is equal to 1 by 4.
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The product of the probability is X1 is equal
to alpha and X2 is equal to 3 alpha which
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is equal to 1 by 4 times half which is equal
to 1 by 8.
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So this is a simple problem.
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Now let us look at a slightly more refined
problem to understand this better.
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Let us look at this example.
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Consider the 4 PAM source, as usual, our 4
PAM source generating IID symbols that this
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independent
identically distributed symbols from the modulation,
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the constellation S is equal to minus 3 alpha,
minus alpha, alpha, and 3 alpha and we set
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the probabilities are 1 by 8, 1 by 8, 1 by
4 and half.
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This is the distribution.
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And we consider a
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block of N is equal to 10 symbols.
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That is we are considering symbols X1, X2,
X3 so on until X9 and X10.
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So basically we are considering a block of
N is equal to 10 symbols.
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And we would like to ask the following questions.
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What is the probability that all the symbols
in the block are elements of the set?
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So the 1st question that we would like to
ask is what is the probability that all the
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symbols in the block belong to the set A equals
minus 3 alpha, alpha.
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So the 1st question.
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So what we are saying is the following thing.
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Understand this example.
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We are considering a source which is generating
IID 4 PAM symbol.
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That is, each 4 PAM symbol, that is each symbol
belongs to the 4-ary PAM constellation, pulse
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amplitude modulation constellation.
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And each symbol is generated independently.
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And it is identically distributed according
to the distribution of the 4 PAM constellation.
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And we are considering a block of 10 such
symbols that is X1, X2, until X10 where each
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of these symbols are generated in an IID fashion.
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And we are asking the question, the 1st question
that we are asking is, what is the probability
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that all of these symbols that is each of
these symbols, X1, X2, until X10 belongs to
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the set A which is minus 3 alpha, alpha.
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That is basically, we are asking the question,
probability that each Xi belongs to the set
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minus 3 alpha, alpha.
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Now let us 1st compute the probability that
a given symbol Xi belongs to capital A. That
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is basically the probability that a given
symbol Xi belongs to the set minus 3 alpha,
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alpha and this is equal to the sum of the
probabilities corresponding to the symbols
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minus 3 alpha and alpha.
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This is basically equal to 1 by 8 plus 1 by
4.
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That is equal to 3 by 8.
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This is something that we have already seen.
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That is, what are we saying?
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We are saying that the probability that a
given symbol Xi belongs to the set A minus
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3 alpha, alpha is 1 by 4 plus 1 by 8 equals
3 by 8.
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Now let us ask the question, what is the probability
that all the symbols belong to this, all the
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symbols that belong to the set A, that is
all the symbols, either minus 3 alpha or alpha,
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that probability is basically the probability,
we are asking the question, what is the probability
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that X1, X2, up to X10 belong to the set A
which is now because these symbols are independent.
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This is now basically the product of the probability
X1 element of A times the probability X2 element
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of A times the probability X10 belongs to
A which can now be seen to be given as, each
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of these probabilities, probability X1 belongs
to A is 3 by 8 times 3 by 8 and so on and
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so forth, the product of 3 by 8 ten times
which is equal to 3 by 8 raised to the power
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10.
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So therefore we have now computed, using the
independence and identical distribution property,
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we have computed the probability that all
of these symbols, X1, X2, up to X10 belongs
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to the set A. That is each of these Xis, X1,
X2, up to X10 is either minus 3 Alpha and
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Alpha.
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The probability of that is basically 3 by
8 raised to the power 10.
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And we have used the independent identical
distribution property of these various 4 PAM
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symbols to calculate that.
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Let us now extend this example further.
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Let us now ask the question, what is the probability
that none of
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the symbols belong to A equals, now we are
asking the reverse question.
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1st we answered the question, what is the
probability that all the symbols belong to
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A?
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Now, what is the probability that none of
the symbols belong to A?
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And this answer is also fairly straightforward.
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If none of the symbols belong to A, then all
the symbols must belong to A complement.
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That is, if none of the symbols are either
minus 3 alpha or alpha, it with all of the
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symbols basically should be either minus 3
alpha or alpha.
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Therefore, the probability that a given symbol
does not belong to A is simply the probability
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that a symbol belongs to A complement.
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That is, the probability that the symbol belongs
to the set minus alpha, 3 alpha.
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And this is also the probability Xi belongs
to A complement is 1 minus the probability
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remember, the probability of A complement
is 1 minus the probability of A. That is equal
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to 1 minus 3 by 8 which is equal to 5 by 8.
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Now we are asking the question, what is the
probability that all of these X1, that is
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none of them belong to A which means all of
them belong to A complement which is basically
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again since X1, X2, up to X10 are independent,
that is simply the product of the probabilities
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X1 belongs to A complement times the probability
X2 belongs to A complement times the probability
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X10 belongs to A complement which is equal
to 5 by 8 times 5 by 8 equals 5 by 8 raised
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to the power 10.
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Therefore now, we have answered the question,
what is the probability that none of the symbols
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belong to A?
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Which means all of the symbols must belong
to A complement and the probability of that
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is 5 by 8 raised to the power of 10.
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All right?
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Now let us ask another question which is the
last question.
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This is part C of the same question which
is basically the following thing.
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Let us ask the question, what is the probability
that at least one the symbols belongs to A?
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All right?
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So we are asking the question that we have
this block of symbols, X1, X2, up to X10.
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What is the probability that at least one
of these X1, X2, up to X10 belongs to the
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set A?
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That is at least one of these X1, X2, up to
X10 is either minus 3 alpha or alpha.
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That is at least 1 of X1, X2, up to X10 is
either minus 3 alpha or alpha and this probability
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is also very simple.
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The probability that at least one belongs
to A is basically 1 minus the probability
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that none of them belong to A.
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And we have already computed in the previous
part what is the probability that none of
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them belong to A?
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And therefore the probability
of at least one of them belonging to A equals
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1 minus the probability that none of them
belong to A which is basically equal to 1
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minus 5 by 8 raised to the power of 10.
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And this is very simple.
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Right?
220
00:24:31,230 --> 00:24:32,230
So we have calculated.
221
00:24:32,230 --> 00:24:36,410
What we are saying is the probability that
at least one of these symbols in this block,
222
00:24:36,410 --> 00:24:44,160
X1, X2, up to X10 belongs to A is basically
1 minus the probability that none of them
223
00:24:44,160 --> 00:24:47,800
belongs to A and the probability that none
of them belongs to A is 5 by 8 raised the
224
00:24:47,800 --> 00:24:48,800
power of 10.
225
00:24:48,800 --> 00:24:54,090
Therefore the probability that one of them
belongs to A is 1 minus 5 by 8 raised to the
226
00:24:54,090 --> 00:24:58,840
power of 10 and that is what we have computed
over here..
227
00:24:58,840 --> 00:25:00,280
So what have we seen in this module?
228
00:25:00,280 --> 00:25:05,270
In this module, we have seen something interesting.
229
00:25:05,270 --> 00:25:11,870
We have seen that the concept of Independence
as applied to the context of digital communication
230
00:25:11,870 --> 00:25:13,750
system or wireless communication systems.
231
00:25:13,750 --> 00:25:20,780
We have said that although, the previous examples,
we had seen that the 2 events were not independent.
232
00:25:20,780 --> 00:25:26,800
When we looked at the 2 different symbols,
the 2 different symbols generated at different
233
00:25:26,800 --> 00:25:33,080
time instant by a communication system, these
are independent.
234
00:25:33,080 --> 00:25:37,720
And therefore, one can calculate, now given
a block of this, we have looked at an example
235
00:25:37,720 --> 00:25:41,460
about the various properties that is given
a block of these symbols that is X1, X2, up
236
00:25:41,460 --> 00:25:46,860
to X10 that is given a block comprising of
10 symbols, what are the various properties?
237
00:25:46,860 --> 00:25:52,210
What can we say about the probability that
none of the symbols belong to a certain set
238
00:25:52,210 --> 00:25:53,210
A?
239
00:25:53,210 --> 00:25:56,180
What is the probability that one of the symbols
belong to a certain set A?
240
00:25:56,180 --> 00:26:05,690
All of these, exploiting basically the independent
identical distributed nature of these various
241
00:26:05,690 --> 00:26:11,850
symbols generated by the source, generated
by the digital communication source or the
242
00:26:11,850 --> 00:26:14,010
source in the wireless communication system.
243
00:26:14,010 --> 00:26:21,200
So this is an important aspect or this is
an important, key impact of we can say the
244
00:26:21,200 --> 00:26:27,240
relevance of this property of independence
or independent event in the context of wireless
245
00:26:27,240 --> 00:26:30,230
communication systems.
246
00:26:30,230 --> 00:26:34,830
So we conclude this module here.
247
00:26:34,830 --> 00:26:39,280
Thank you very much.