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p. 1
TELE3113 Analogue & Digital Communications
Review of Probability Theory
p. 2
Probability and Random Variables
Concept of Probability:
When the outcome of an event is not always the same, probability is the measure of the chance of obtaining a particular possible outcome
outcomeslikely equally possible ofnumber totaloutcomesfavourable possibleofnumber
outcomes favourable =)(P
NNAP A
N ∞→= lim)( Where N is total number of event occurrence,
NA is the number of occurrence of outcome A
e.g. dice tossing: P2 = 1/6 ; P2 or 4 or 6 = 1/2
p. 3
Common Properties of Probability
• 0≤ P(A) ≤ 1
• If there are N possible outcomes A1 , A2 , … , AN then
• Conditional Probability:
probability of the outcome of an event is conditional on the outcome of another event
∑=
=N
iiAP
11)(
P(A)B)andP(AA)|P(B =
P(B)B)andP(AB)|P(A =;
P(B)A)P(BAPB)P(A |)(| =Bayes’ theorem
p. 4
Common Properties of Probability
• Mutually exclusiveness
P(A or B) = P(A) + P(B); P(A and B) = 0
• Statistically Independence
P(B)P(A) B)and P(A(B)| P(A (A)|P(B
⋅=⇒== );) APBP
P(B)B)andP(AB)| P(A;
P(A)B)andP(AA)|P(B ==Q
Thus, A and B are statistically independent.
Thus, A and B are mutually exclusive.
p. 5
Communication Example
In a communication channel, signal may be corrupted by noises.
m0
m1
r0
r1
P(r0|m0)
P(r0|m1)P(r1|m0)
P(r1|m1)
If r0 is received, m0 should be chosen ifP(m0|r0)P(r0) > P(m1|r0)P(r0)
By Bayes’ theorem P(r0|m0)P(m0) > P(r0|m1)P(m1)
Similarly, if r1 is received, m1 should be chosen ifP(m1|r1)P(r1) > P(m0|r1)P(r1) ⇒ P(r1|m1)P(m1) > P(r1|m0)P(m0)
Probability of correct reception: P(c) = P(ro|mo)P(mo) + P(r1|m1)P(m1)
Probability of error: P(ε) = 1-P(c)
p. 6
Random Variables
A random variable X(.) is the rule or functional relationship which assigns real numbers X(λi) to each possible outcome λi in an experiment.
For example, in coin tossing, we can assign X(head) = 1, X(tail) = -1
If X(λ) assumes a finite number of distinct values
discrete random variable
If X(λ) assumes any values within an interval
continuous random variable
p. 7
Cumulative Distribution Function
The cumulative distribution function, FX(x) , associated with a random variable X is:
)( xXPxFX ≤=Properties:
•
•
•
•
1)(0 ≤≤ xFX1)(;0)( =∞=−∞ FF
2121 )()( xxxFxF ≤≤ if
)()( 1221 xFxFxXxP XX −=≤<
FX(x)
x0
1
FX(x)
x0
1
(Non-decreasing)
p. 8
Probability Density Function
The probability density function, fX(x) , associated with a random variable X is:
dxxdFxf X
X)(
)( =
∫∞−
==≤x
XX dfxFxXP ββ )()(
∫∞
∞−
= 1)( dxxf X
∫=−=≤<2
1
)()()( 1221
x
xXXX dxxfxFxFxXxP
0)( ≥xf X
Properties:
•
•
•
•
FX(x)
x0
1
fX(x)
x0
for all x
p. 9
Statistical Averages of Random Variables
The statistical average or expected value of a random variable X is defined as
Xi
ii mxPxXE == ∑ )(
EX is called the first moment of X and mX is the average or mean value of X.
Similarly, the second moment EX2 is
∑=i
ii xPxXE )( 22
Its square root is called the root-mean-square (rms) value of X.
XmdxxxfXE == ∫∞
∞−
)(
∫∞
∞−
= dxxfxXE )( 22
or
or
The variance of the random variable X is defined as
222222 )()()( XXXXX mXEdxxfmXmXE −=−=−= ∫∞
∞−
σσ or
The square root of the variance is called the standard deviation, σX, of the random variable X.
p. 10
Statistical Averages of Random Variables
∑∑==
=
N
iii
N
iii XEaXaE
11
∑∑==
=
N
iii
N
iii XVaraXaVar
1
2
1
Expected value of linear combination of N random variables is equivalent to linear combination of expected values of individual random variables
For N statistically independent random variables: X1, X2, … , XN
Covariance of a pair of random variables: X, Y
YXYXXY mmXYEmYmXE −=−−= ))((µ
If X and Y are statistically independent, µXY=0
p. 11
A random variable that is equally likely to take on any value within a given range is said to be uniformly distributed.
Uniform Distribution
p. 12
Consider an experiment having only two possible outcomes, A and B, which are mutually exclusive.
Let the probabilities be P(A) = p and P(B) = 1 − p = q.
The experiment is repeated n times and the probability of A occurring itimes is ,)( ini qp
in
iAP −
== where (binomial coefficient).
)!(!!ini
nin
−=
Binomial Distribution
The mean value of the binomial distribution is np and the variance is (npq).
p. 13
Gaussian Distribution
Central-Limit theorem: The sum of N independent, identically distributedrandom variables approaches a Gaussian distribution when N is very large.
22 2/)(
21)( σµ
σπ−−= xexf
The Gaussian pdf is continuous and is defined by
where µ is the mean , and σ2 is the variance .
cumulative distribution function:
∫∞−
−−=
≤=x
y
X
dye
xXPxF22 2/)(
21
)(
σµ
σπ
p. 14
Gaussian Distribution
Zero-mean unit-variance Gaussian random variable: 2/2
21)( xexg −=π
⇒ Probability distribution function: ∫∫∞−
−
∞−
==Ωx
yx
dyedyygx 2/2
21)()(π
Define Q-function: ∫∞
−=Ω−=x
y dyexxQ 2/2
21)(1)(π
(monotonic decreasing)
In general, for a random variable X with pdf:22 2/)(
21)( σµ
σπ−−= xexf
−
=>
−
Ω=≤σµ
σµ xQxXPxxXP )()( ;
Define: error function (erf) and complementary error function (erfc) :
=
+=Ω
221
21)( xerfcxerfx
21Q(x) ;
∫∫∞
−− =−==x
yx
y dyexerfxerfcdyexerf22 2)(1)(2)(
0 ππ ;
Thus,
p. 15
Q-function
∫∞
−=x
y dyexQ 2/2
21)(π
1for x >>≅−
π2)(
2/2
xexQ
x
p. 16
Random Processes
A random process is a set of indexed random variables (sample functions) defined in the same probability space.
In communications, the index is usually in terms of time.
xi(t) is called a sample functionof the sample space.
The set of all possible sample functions xi(t) is called ensemble and defines the random process X(t).
For a specific i, xi(t) is a time function.For a specific ti, X(ti) denotes a random variable.
p. 17
Random Processes: Properties
Consider a random process X(t) , let X(tk) denote the random variable obtained by observing the process X(t) at time tk .
Mean: mX(tk) =EX(tk)
Variance:σX2 (tk) =EX2(tk)-[mX (tk)] 2
Autocorrelation: RXtk ,tj=EX(tk)X(tj) for any tk and tj
Autocovariance: CXtk ,tj=E [X(tk)- mX (tk)][X(tj)- mX (tj)]
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