Download Random Variables and Stochastic Processes * 0903720 Lecture#18

Random Variables and Stochastic
Processes – 0903720
Lecture#18
Dr. Ghazi Al Sukkar
Email: [email protected]
Office Hours: Refer to the website
Course Website:
http://www2.ju.edu.jo/sites/academic/ghazi.alsukkar
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Chapter 9
Stochastic Processes
 Introduction
 Basic Definitions
 Statistical properties of Random Process
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Introduction
• Signals can be classified into two main groups:
– Deterministic
– Random
• Random signals can be described by properties e.g.
1. Average power.
2. Spectral distribution on the average.
3. The probability that the signal amplitude exceeds a given
value.
• The probabilistic models used to describe random signals are
called a random process (stochastic process or time series).
• Examples of random processes in communications:
– Channel noise,
– Information generated by a source,
– Interference.
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Definition
• Recall that a RANDOM VARIABLE 𝑋(𝜁), is a rule for
assigning to every outcome 𝜁𝑖 of an experiment with a
sample space 𝑆 a number 𝑋(𝜁𝑖 ).
– Note: 𝑋 denotes a random variable and 𝑋(𝜁𝑖 ) denotes a
particular value of it.
• A RANDOM PROCESS 𝑋(𝑡, 𝜁) is a rule for assigning to every
outcome 𝜁𝑖 a waveform (function of time) 𝑋(𝑡, 𝜁𝑖 ).
– Note: for notational simplicity we often omit the
dependence on 𝜁𝑖 .
⟹ 𝑋(𝑡) denotes a Random process.
• Thus a stochastic process is a family (ensemble) of time
functions depending on the parameter 𝜁 or, equivalently, a
function of 𝑡 and 𝜁.
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The set of all possible functions is called the
ENSEMBLE 𝑋(𝑡, 𝜁)
𝜁1
𝜁2
𝜁𝑛
𝑋(𝑡, 𝜁1 )
Sample function
(realization)
Sample function 𝑋(𝑡, 𝜁2 )
(realization)
Sample function 𝑋(𝑡, 𝜁𝑛 )
(realization)
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Example
• At time 𝑡 = 0 a die is tossed, a time function 𝑥𝑖 (𝑡) is
assigned to each possible outcome of the experiment:
Outcome
Waveform
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𝑥1 𝑡 = −4
2
𝑥2 𝑡 = −2
3
𝑥3 𝑡 = 2
4
𝑥4 𝑡 = 4
5
𝑥5 𝑡 = −𝑡/2
6
𝑥6 𝑡 = 𝑡/2
• Then 𝑋 is a Random Process:
outcome of experiment→ set of waveforms
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• A random process is denoted by: 𝑋(𝑡, 𝜁)
where 𝑡 represents time, and 𝜁 is a variable that represents an
outcome in the sample space 𝑆.
• 𝑋(𝑡, 𝜁) can denote the following quantities:
1)
𝑋 𝑡, 𝜁𝑖 = 𝑥𝑖 (𝑡) a specific member function (sample function), in
2)
𝑋 𝑡, 𝜁 = 𝑋 𝑡, 𝜁𝑖 |𝜁𝑖 ∈ 𝑆 = 𝑥1 𝑡 , 𝑥2 𝑡 , … a collection (ensemble)
this case, 𝑡 is variable a 𝜁 is fixed.
of time functions (Stochastic process), in this case both 𝑡 and 𝜁 are
variables.
3) 𝑋 𝑡𝑜 , 𝜁 = 𝑋 𝑡𝑜 , 𝜁𝑖 |𝜁𝑖 ∈ 𝑆 = 𝑥1 𝑡𝑜 , 𝑥2 𝑡𝑜 , … , a
collection of numerical values, which represent a R.V. equal
to the state of the given process at 𝑡𝑜 . In this case 𝑡 is fixed and
𝜁 is variable.
4) 𝑋 𝑡𝑜 , 𝜁𝑖 = 𝑥𝑖 (𝑡𝑜 ) which is a number represent the value of the
sample function 𝑥𝑖 (𝑡) at 𝑡𝑜 .
• Instead of using the above notations, 𝑋(𝑡) is used to denote
all of them. Usually the meaning can be understood from the
context.
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𝑋(𝑡, 𝜁)
𝑋(𝑡, 𝜁2 )
𝑋(𝑡, 𝜁3 )
𝑥
𝑋(𝑡, 𝜁1 )
0
𝑡
𝑋(𝑡, 𝜁4 )
t
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑋 𝑡, 𝜁1 =?
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑋 𝑡, 𝜁1 = 𝑋 𝑡, 𝜁 = 1 = 𝑥1 𝑡 = −4
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑋 𝑡, 𝜁5 =?
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑋 𝑡, 𝜁5 = 𝑋 𝑡, 𝜁 = 5 = 𝑥5 𝑡 = −𝑡/2
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑋 6, 𝜁 =?
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑋 6, 𝜁 = 𝑋 6 which is a R.V. with the values −4, −3, −2, 2, 3, 4
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• Therefore, a general Random or Stochastic Process can be
described as:
– Collection of time functions (signals) corresponding to various
outcomes of random experiments.
– Collection of random variables observed at different times.
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Regular vs. Predictable S.P.
• Regular Stochastic Process: consists of a family of functions
that can not be described in terms of a finite number of
parameters. Furthermore, the future of a sample 𝑋(𝑡, 𝜁) of
𝑋(𝑡) can not be determined in terms of its past.
– Brownian Motion
• Motion of all particles (ensemble)
• Motion of a specific particle (sample function)
• Predictable Stochastic Process: consist of a family of functions
that can be described in terms of finite number of
parameters.
– Voltage of a generator with fixed frequency
• Amplitude and phase are random variables
𝑉 𝑡 = 𝑅 cos(𝜔𝑡 + 𝜙), where 𝑅 𝑎𝑛𝑑 𝜙 are R.V.s
⟹ 𝑉 𝑡, 𝜁𝑖 = 𝑅 𝜁𝑖 cos 𝜔𝑡 + 𝜙(𝜁𝑖 )
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Classification of Random processes
Continuous
𝑿(𝒕)
Discrete
𝒕
Continuous
Continuous-time and
Continuous-state Process
Discrete-time and
Continuous-state Process
(Continuous-state
sequence)
Discrete
Continuous-time and
Discrete-state Process
Discrete-time and
Discrete-state Process
(Discrete-state sequence)
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Equality
• Equality (Generally)
– Ensembles should be equal for each “𝜁” and “𝑡”
𝑋 𝑡, 𝜁 = 𝑌(𝑡, 𝜁)
• Equality (Mean Square Sense)
– If the following equality holds
𝐸 𝑋 𝑡 − 𝑌(𝑡)
– Sufficient in many applications
2
=0
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Statistics of Stochastic Process
• A stochastic process 𝑋(𝑡) is a non-countable infinity of
random variables, one for each 𝑡.
• For fixed 𝑡, 𝑋(𝑡) represents a random variable:
 Its First-Order distribution function is given by:
FX  x, t   PX t   x
It depends on 𝑡, since for a different 𝑡, we obtain a
different random variable, and it is the probability of the
event 𝑋(𝑡) ≤ 𝑥 which consist of all outcomes 𝜁 such
that, at specific time 𝑡, the samples 𝑋(𝑡, 𝜁) does not
exceed the number 𝑥.
 Its First-Order PDF is: f x, t    F x, t 
X
x
X
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For this figure 𝐹𝑋 𝑥, 𝑡 = 𝑃 𝑋(𝑡) ≤ 𝑥 = 𝑃 𝜁3 , 𝜁4
X (t ,  )
𝑋(𝑡, 𝜁2 )
𝑋(𝑡, 𝜁3 )
𝑥
𝑋(𝑡, 𝜁1 )
0
𝑡
𝑋(𝑡, 𝜁4 )
t
In general if 𝐴 = 𝜁𝑖 |𝑋(𝑡, 𝜁𝑖 ) ≤ 𝑥 , then 𝐹𝑋 𝑥, 𝑡 = 𝑃(𝐴)
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑃(𝑋 4 = −2) =?
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
1
𝑃 𝑋 4 = −2 = 𝑃 2,5 =
3
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑃(𝑋 4 ≤ 0) =?
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑃 𝑋 4 ≤ 0 = 𝑃 1,2,5 = 1/2
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• Second-Order CDF of a random process:
For 𝑡 = 𝑡1 and 𝑡 = 𝑡2, 𝑋(𝑡) represents two different
random variables 𝑋1 = 𝑋(𝑡1) and 𝑋2 = 𝑋(𝑡2) respectively.
Their joint distribution is given by
FX ( x1 , x 2 ; t1 , t 2 )  P{ X (t1 )  x1 , X (t 2 )  x 2 }
• Second-Order PDF of a random
process:
2
f X  x1 , x 2 ; t1 , t 2  

FX  x1 , x 2 ; t1 , t 2 
x1 .x 2
X (t ,  )
𝑥2
𝑥1
𝑋(𝑡, 𝜁2 )
𝑋(𝑡, 𝜁3 )
𝑋(𝑡, 𝜁1 )
0
t1
t2
𝑋(𝑡, 𝜁4 )
t
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑃(𝑋 0 = 0, 𝑋 4 = −2) =?
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑃 𝑋 0 = 0, 𝑋 4 = −2 = 𝑃 5
= 1/6
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑃 𝑋 4 = −2|𝑋 0 = 0 =?
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𝑆 = 1,2,3,4,5,6 = 𝜁1 , 𝜁2 , 𝜁3 , 𝜁4 , 𝜁5 , 𝜁6
𝑃 𝑋 4 = −2|𝑋 0 = 0 =
𝑃(𝑋 4 = −2, 𝑋 0 = 0) 1/6
=
= 1/2
𝑃(𝑋 0 = 0)
2/6
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• Relation between first-order and second-order can be presented as
𝐹𝑋 𝑥1 ; 𝑡1 = 𝐹𝑋 𝑥1 , ∞; 𝑡1 , 𝑡2
∞
𝑓𝑋 𝑥1 ; 𝑡1 =
𝑓𝑋 𝑥1 , 𝑥2 ; 𝑡1 , 𝑡2 𝑑𝑥2
−∞
• The nth order distribution of 𝑋(𝑡) is defined as the joint distribution:
FX ( x1 , x 2 ,  x n ; t1 , t 2 , t n )  PX (t1 )  x1 , X (t 2 )  x 2 ,..., X (t n )  x n 
•
•
A stochastic process is defined to be completely or totally
characterized if the joint densities for the random variables
𝑋 𝑡1 , 𝑋 𝑡2 , … , 𝑋(𝑡𝑛 ) are known for all times 𝑡1 , 𝑡2 , … , 𝑡𝑛 and all
𝑛.
In general, a complete characterization is practically impossible,
except in rare cases. As a result, it is desirable to define and work
with various partial characterizations. Depending on the
objectives of applications, a partial characterization often suffices
to ensure the desired outputs.
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