Download Chapter 8 Probability and Random variables 1

Chapter 8
Probability and Random variables
F. G. Stremler, Introduction to Communication Systems 3/e
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NA
P( A)  lim
N  N
• Probability
• All possible outcomes (A1 to AN) are included
N
 P( A )  1
i 1
i
N AB
• Joint probability
P( AB)  lim
N  N
P( AB)  P( B | A) P( A)  P( A | B ) P( B )
• Conditional probability
N AB N AB N P( AB)
N AB N AB N P( AB)
P( A | B ) 


P( B | A) 


NB
NB / N
P( B )
NA
NA / N
P( A)
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• Bayes’ theorem
P( B ) P( A | B )
P( B | A) 
P( A)
• Random 2/52 playing cards. After looking at the
first card, P(2nd is heart)=? if 1st is or isn’t heart
A: a heart on the 1st; B: a heart on the 2nd; C: no heart on the 1st
P(B|A) = 12/51; P(B|C) = 13|51
• Probability of two mutually exclusive events
P(A+B)=P(A)+P(B)
• If the events are not mutually exclusive
P(A+B)=P(A)+P(B)-P(AB)
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Random variables
• A real valued random variable is a real-value
function defined on the events of the probability
system.
• Cumulative distribution function (CDF) of x is
nx  a
F (a)  P( x  a)  lim (
)
n 
n
• Properties of F(a)
• Nondecreasing,
• 0≤F(a)≤1,
F ( )  0
F ( )  1
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Probability density function (PDF)



f x (a)dx  PX  a
F (a)  P( x  a)
dF ( a )
f ( x) 
|a  x
da
Properties of PDF
f ( x )  0.



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f ( x)dx  F ()  1
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Tutorial Q.2
• Consider the experiment that consists in the rolling of two
honest dice. The random variable X is assigned to the
sum of the numbers showing up to the two dice.
Determine and plot the cumulative distribution function
(CDF) and the probability distribution function (pdf) of X.
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Discrete and continuous distributions
• Discrete: random variable has M discrete values
CDF or F(a) was discontinuous as a increase.
Digital communications
M
PDF f ( x )  P( x ) ( x  x )

i 1
i
i
M is the number of discretely events
L
CDF
F ( a )   P( xi )
i 1
L is the largest integer such that xL  a, L  M
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• Continuous distributions: if a random variable is allowed
to take on any value in some interval.
CDF and PDF would be continuous functions.
Analogue communications, noise.
• Expected value of a discretely distributed random
variable
M
y  [h( x )]   h( xi ) P( xi )
i 1
Normalized average power
P=

y2i p(yi)
i
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example
A discrete random signal, y(t), can take one of the
four predefined voltage levels, y1 = 0.5 V, y2 =
0.4 V, y3 = 0.2 V, and y4 = 0.1 V. Assume that
these levels occur with probabilities, p(y1) =
0.2, p(y2) = 0.3, p(y3) = 0.1, and p(y4) = 0.4.
Calculate the average power delivered by y(t) into
a 100Ω resistive load.
The normalized average power, P, is given by:
P=

y2i p(yi)
i
i.e. (0.5)2  0.2 + (0.4)2  0.3 + (0.2)2  0.1 + (0.1)2  0.4 = 0.106/100 W = 1.06 mW
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Important distributions
• Binomial : discrete
• Poisson: discrete
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• Uniform: continuous, a random variable that is
equally likely to take on any value within a
given range.
• Gaussian (normal): continuous
• Normalised Gaussian pdf having zero mean and
unit variance
1
p ( x) 
2
e
 x2 / 2
• Sinusoidal: continuous
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