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Random Variables And Process
Prof. B B Tiwari
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Introduction
Signals may be deterministic or random.
If uncertainty exists then signals are random
signals.
They are not predictable neither are they
completely unpredictable.
The probability of being correct can bepredicted up to certain extent.
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Probability
When the possible outcomes of an experiment isnot always same we deal it with probabilitytheory.
For ex: when an experiment is repeated N timesand the possible outcomes A occurs NAThe relative frequency of occurrence of A is
NA /N .
It can be written asP(A)=
.
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Totally independent events A1 and A2 with
probabilities P(A1) and P(A2) are mutually
exclusive events
P(A1 or A2) = P(A1) + P(A2)
In general
P(A1 or A2or .or AL) = ()
=1 and we know that
=1 = 1
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If there are two events A and B one of the
events may affect the other .In this case the
conditional probability
P(B|A) = P(A|B)*P(B)/P(A)
This result is know as Bayes theorem.
If A and B are totally independent thenP(B|A) = P(B) and
Joint probability P(A,B) = P(A)*P(B)
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Cumulative Distribution Function
F(x) =
and probability density functionf(x)=
F(x)
PDF has the following properties
f(x)>= 0 for all x
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Probability of outcome X being less then or equalto x1 is
P(X
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P(x1
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For two random variables X and Y the
probability that x
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FXY(x,y) = P(X
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A communication example
We want to transmit one of two possible
messages the message m0 that the bit 0 is
intended or the message m1 that the bit 1 is
intended.
When received , generates some voltage , say
r0 , which may be as simple as a dc voltage,
while m1 received generates a voltage r1.
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P(r0|m0)=probability that r0 is received given
that m0 is sent,
P(r1
|m0
)=probability that r1
is received given
that m0 is sent,
P(r=|m1)=probability that r0 is received given
that m1 is sent,
P(r1|m1)=probability that r1 is received given
that m1 is sent,
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Clearly if P(m0|r0)>p(m1|r0) then we shoulddecide that m0 is intended and if the inequality isreversed we should decide for m1.Altogetherthen our algorithm should be :
If r0 is received: Choose m0 if P(m0|r0)>P(m1|r0)
Choose m1 if P(m1|r0)>P(m0|r0)
If r1 is received: Choose m= if P(m0|r1)>P(m1|r1)
Choose m1 if P(m1|r1)>P(m0|r1)
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A receiver which operates in accordance with
this algorithm is said to maximize the
posteriori probability (m.a.p) of a correct
decision and is called an optimum receiver .
P(r0|m0)P(m0)>P(r0|m1)P(m1)
P(r1|m1)P(m1)>P(r1|m0)P(m0)
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Average value of random variable
The possible numerical value of the random
variable X are x1 ,x2x3,., with probabilities of
occurrence P(x1), P(x1), P(x1).
x1P(x1)N + x2P(x2)N+.=N () The mean or average value of all these
measurements and hence the average value of the
random variable is calculated by dividing the sumshown above by the number of measurements N.
X= E(X)=m= ()
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Tchebycheffs Inequality
P(|X|>=)
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Variance of a random variable
2 = E[(X-m)2] = 2
Writing (x-m)2 = x2 -2mx + m2 in the integral ofabove equation and integrating term by term ,
2 = E(X2) - 2m2 + m2
= E(X2) m2
The quantity itself is called the standard
deviation and is the root mean square (rms) valueof (X-m). If the average value m=0, then
2 = E(X2)
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Since x2 >=0 and f(x) >= 0 for all x we have that
2 0
Can be written as2 >= 2 2
+
In the ranges -
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Gaussian Probability Density
F(x)=1
/
X =
/
dx = m
E[(X-m)2] =
/
dx = 2
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Error Function