Chapter: Mathematics (maths) - Random Processes

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Random Processes

Random Processes: Introduction, Classification, stationary processes, Markov processes, Poisson processes, Discrete parameter Markov chains, Chapman Kolmogorov Equation, Limiting distribution, Random Telegraph processes.


RANDOM PROCESSES

·        Introduction

·        Classification

·        stationary processes

·        Markov processes

·        Poisson processes

·        Discrete parameter Markov chains

·        Chapman Kolmogorov Equation

·        Limiting distribution

·        Random Telegraph processes

 

Introduction

 

In previews pages, we discussed about random variables. Random variable is a function of the possible outcomes of a experiment. But, it does not include the concept of time. In the real situations, we come across so many time varying functions which are random in nature. In electrical and electronics engineering, we studied about signals.

 

Generally, signals are classified into two types.

(i)  Deterministic

 

(ii)  Random

 

Here both deterministic and random signals are functions of time. Hence it is possible for us to determine the value of a signal at any given time. But this is not possible in the case of a random signal, since uncertainty of some element is always associated with it. The probability model used for characterizing a random signal is called a random process or stochastic process.

 

RANDOM PROCESS CONCEPT

 

A random process is a collection (ensemble) of real variable {X(s, t)} that are functions of a real variable t where s  S, S is the sample space and t T. (T is an index set).

 

 

REMARK

i) If t is fixed, then {X(s, t)} is a random variable.

ii) If S and t are fixed {X(s, t)} is a number.

 

iii) If S is fixed, {X(s, t)} is a signal time function.

 

NOTATION

 

Here after we denote the random process {X(s, t)} by {X(t)} where the index set T is assumed to be continuous process is denoted by {X(n)} or {Xn}.

 

A comparison between random variable and random process

 


 

Random Variable

 

A function of the possible outcomes of an experiment is X(s)

Outcome is mapped into a number x.

 

Random Process

 

A function of the possible outcomes of an experiment and also time i.e, X(s, t)

Outcomes are mapped into wave from which is a fun of time 't'.

 

 

 

CLASSIFICATION OF RANDOM PROCESSES

We can classify the random process according to the characteristics of time t and the random variable X = X(t) t & x have values in the ranges –< t < and –< x <.


 

1 CONTINUOUS RANDOM PROCESS

If 'S' is continuous and t takes any value, then X(t) is a continuous random variable.

Example

 

Let X(t) = Maximum temperature of a particular place in (0, t). Here 'S' is a continuous set and t  0 (takes all values), {X(t)} is a continuous random process.

 

2 DISCRETE RANDOM PROCESS

If 'S' assumes only discrete values and t is continuous then we call such random process {X(t) as Discrete Random Process.

Example

 

Let X(t) be the number of telephone calls received in the interval (0, t). Here, S = {1, 2, 3, …}

 

T = {t, t  0}

 

 {X(t)} is a discrete random process.

 

3 CONTINUOUS RANDOM SEQUENCE

If 'S' is a continuous but time 't' takes only discrete is called discrete random sequence. Example: Let Xn denote the outcome of the nth toss of a fair die.

 

Here  S = {1, 2, 3, 4, 5, 6} T = {1, 2, 3, …}

 

 (Xn, n = 1, 2, 3, …} is a discrete random sequence.

 

CLASSIFICATION  OF  RANDOM  PROCESSES  BASED  ON  ITS  SAMPLE

 

FUNCTIONS Non-Deterministic Process

 

A Process is called non-deterministic process if the future values of any sample function cannot be predicted exactly from observed values.

 

Deterministic Process

 

A process is called deterministic if future value of any sample function can be predicted from past values.

 

1 STATIONARY PROCESS

 

A random process is said to be stationary if its mean, variance, moments etc are constant. Other processes are called non stationary.

 

1. 1st Order Distribution Function of {X(t)}

For a specific t, X(t) is a random variable as it was observed earlier.

 

F(x, t) = P{X(t)  x} is called the first order distribution of the process {X(t)}.

 

1st Order Density Function of {X(t)}


 

2nd Order distribution function of {X(t)}

(1 , x 2 ; t 1 , t 2 ) = P {(1 )  x 1 ; X (2 )  x2 }is the point distribution of the random variables X(t1) and X(t2) and is called the second order distribution of the process {X(t)}.

2nd order density function of {X(T)}


 

 

2 First - Order Stationary Process

Definition

A random process is called stationary to order, one or first order stationary if its 1st order density function does not change with a shift in time origin.

In other words,

 

X (1 , t 1 ) = f X (1 , t 1 + C)must be true for any t1 and any real number C if {X(t1)} is to be a first order stationary process.

 

Example :3.3.1

Show that a first order stationary process has a constant mean.

Solution

Let us consider a random process {X(t1)} at two different times t1 and t2.


Mean process {X(t1)} = mean of the random process {X(t2)}.

Definition 2:

If the process is first order stationary, then Mean = E(X(t)] = constant

 

4 Second Order Stationary Process

 

A random process is said to be second order stationary, if the second order density function stationary.

(1 , x 2 ; t 1 , t 2 ) = f (1 , x 2 ; t 1 + C, t 2 + C )1 , x2 and C.

(X12 ), E (22 ), E (X1 , X2 )denote change with time, where

 

X = X(t1); X2 = X(t2).

 

5 Strongly Stationary Process

A   random process is called a strongly stationary process or Strict Sense Stationary

 

Process (SSS Process) if all its finite dimensional distribution are invariance under translation of time 't'.

fX(x1, x2; t1, t2) = fX(x1, x2; t1+C, t2+C)

 

fX(x1, x2, x3; t1, t2, t3) = fX(x1, x2, x3; t1+C, t2+C, t3+C) In general

fX(x1, x2..xn; t1, t2…tn) = fX(x1, x2..xn; t1+C, t2+C..tn+C) for any t1 and any real number C.

 

6 Jointly - Stationary in the Strict Sense

{X(t)} and Y{(t)} are said to be jointly stationary in the strict sense, if the joint distribution of X(t) and Y(t) are invariant under translation of time.

Definition Mean:


 

7Auto Correlation of a Random Process

Let X(t1) and X(t2) be the two given numbers of the random process {X(t)}. The auto correlation is


 

8 Auto Covariance of A Random Process


 

Where CXX (t1, t2) denotes the auto covariance.

 

 

 

 


FIRST ORDER STRICTLY STATIONARY PROCESS

 

Stationary Process (or) Strictly Stationary Process (or) Strict Sense Stationary Process [SSS Process]

 

A random process X(t) is said to be stationary in the strict sense, it its statistical characteristics do not change with time.

Stationary Process:


1) Consider the RP X(t) = Cos (w0t +θ) where θis uniformly distributed in the interval -π to π. Check whether X(t) is stationary or not? Find the first and Second moments of the process.

 

Given X(t) = cos (W0t + θ)

 

Where θis uniformly distributed in (-π, π)


To prove              (i) X(t) is SSS process

                   (ii) E[X(t)] = Constant

                   (iii) Var [X(t)] = Constant


S.T the RP X(t): Acos (w 0t is not stationary if A and w0 are constants and θis

 

uniformly distributed random variable in (0, π). In X(t) = Acos (w 0t

 

In 'θ'uniformly distributed


N(t) is not a stationary process.

 

SECOND ORDER AND WIDE SENSE STATIONARY PROCESS

 

A Process is said to be second order stationary, if the second order density function statistics.


If a random process X(t) is WSS then it must also be covariance stationary. In X(t) is WSS

i) E[X(t)] = µ= a const.

 

(ii) R(t1, t2) = a fy of (t 1 - t2) The auto covariance fn is gn by


Which depends only on the time difference. Hence X(t) is covariance stationary.

If X(t) is a wide sense stationary process with auto correlation R ()τ =Ae−α(),

determine the second order moment of the random variable X(8) - X(5).


The second moment of X(8) - X(5) is given by


= A + A - 2Ae-3α

 

= 2A(1–e–3α







CROSS CORRELATION

 

The cross correlation of the two random process {X(t)} and {Y(t)} is defined by RXY (t1, t2) = E[X(t1) Y (t2)]

 

 

 

WIDE - SENSE STATIONARY (WSS)

 

A random process {X(t)} is called a weakly stationary process or covariance stationary process or wide-sense stationary process if

i) E{X(t)} = Constant

 

ii) E[X(t) X(t+τ] = Rxx(τ) depend only on τ when τ = t2 - t1.

 

REMARKS :

 

SSS        Process of order two is a WSS Process and not conversely.

 

 

EVOLUTIONARY PROCESS

 

A random process that is not stationary in any sense is called as evolutionary process.

 

SOLVED PROBLEMS ON WIDE SENSE STATIONARY PROCESS

Example:3.6.1

Given an example of stationary random process and justify your claim.

Solution:

 

Let us consider a random process X(t) = A as (wt + θ) where A &ω are custom and 'θ' is uniformlydistribution random Variable in the interval (0, 2π).

 

Since 'θ' is uniformly distributed in (0, 2π), we have

 



Hence Poisson process is not a stationary process.

 

 ERGODIC PROCESS

Ergodic Process are processes for which time and ensemble (statistical) averages are interchangeable the concept of ergodicity deals with the equality of time and statistical average.

 

Time Average


 

 

MARKOV PROCESS - MARKOV CHAINS

1 Markov Process

          A random process X(t) is said to be Markov Process, it


 

2 Markov Chain

          A discrete parameter Markov Process is called Markov Chain.




Example :3.4.1

An Engineering analysing a series of digital signals generated by a testing system observes that only 1 out of 15 highly distorted signals followed a highly distorted signal with no recognizable signal, where as 20 out of 23 recognizable signals follow recognizable signals with no highly distorted signals b/w. Given that only highly distorted signals are not recognizable. Find the fraction of signals that are highly distorted.

 

π1= The fraction of signals that are recognizable [R]

π2 = The fraction of signals that are highly distorted [D]









TYPE 5

 

A training process is considered as two State Markov Chain. If it rain, it is considered to be state 0 & if it does not rain the chain is in stable 1. The tmp of the Markov Chain is defined as


i.         Find the Prob. That it will rain for 3 days from today assuming that it is raining

today.

ii. Find also the unconditional prob. That it will rain after 3 days with the initial Prob. Of state ) and state 1 as 0.4 & 0.6 respectively.




Three boys A, B, C are throwing a ball each other. A always throws the ball to B & B always throws the ball to C but C is just as like to throw the ball to B as to A. State that the process is Markov Chain. Find the tpm and classify the status.



ERGODIC RANDOM PROCESS

 

Time Average

 

The time average of a random process {X(t)} is defined as


Ensemble Average

The ensemble average of a random process {X(t)} is the expected value of the random variable X at time t

Ensemble Average = E[X(t)]

Ergodic Random Process

{X(t)} is said to be mean Ergodic


 

Mean Ergodic Theorem

 

Let {X(t)} be a random process with constant mean µ and let XT be its time average. Then {X(t)} is mean ergodic if


 

Correlation Ergodic Process

The stationary process {X(t)} is said to be correlation ergodic if the process {Y(t)} is mean ergodic where

 

Y(t) = X(t) X(t+λ)

 


 

 

 

MARKOV PROCESS 

Definition

A random process {X(t)} is said to be markovian if


 

Examples of Markov Process

1.The probability of raining today depends only on previous weather conditions existed for the last two days and not on past weather conditions.

2.A different equation is markovian.

Classification of Markov Process


 

 

MARKOV CHAIN

Definition

 

We define the Markov Chain as follows

 

If P{Xn = an/Xn-1 = an-1, Xn-2 = an-2, … X0 = a0}

 

P{Xn = an / Xn-1 = an-1} for all n. the process {Xn}, n = 0, 1, 2… is called as Markov Chains.

 

1.a1, a2, a3, … an are called the states of the Markov Chain.

2.The conditional probability P{X n = aj | X n 1 = ai} = Pij (1, n) is called the one step

 

transition probability from state a i to state a j at the nth step. 3.The tmp of a Markov chain is a stochastic matricx

 

i) Pij 0

 

ii) ΣPij = 1   [Sum of elements of any row is 1]

 

 

 

 

Poisson Process

The Poisson Process is a continuous parameter discrete state process which is very useful model for many practical situations. It describe number of times occurred. When an experiment is conducted as a function of time.

 

Property Law for the Poisson Process

 

Let λ be the rate of occurrences or number of occurrences per unit time and Pn(t) be the probability of n occurrences of the event in the interval (0, t) is a Poisson distribution with parameter λt.


 Second Order Probability Function of a Homogeneous Poisson Process


 

 

SEMI RANDOM TELEGRAPH SIGNAL PROCESS

If N(t) represents the number of occurrence of a specified event in (0, t) and X(t) = (–)N(t), then {X(t)} is called a semi-random telegraph signal process.

 

1 RANDOM TELEGRAPH SIGNAL PROCESS

 

Definition

A random telegraph process is a discrete random process X(t) satisfying the following:

 

i.       X(t) assumes only one of the two possible values 1 or –1 at any time 't'

 

ii.     X(0) = 1 or –1 with equal probability 1/2

 

iii.  The number of occurrence N(t) from one value to another occurring in any interval of length 't' is a Poisson process with rate λ, so that the probability of exactly 'r' transitions is

 


Note: The process is an example for a discrete random process.

* Mean and Auto Correlation P{X(t) = 1} and P{X(t) = 1" for any t.

 

 



POISSON BINOMIAL PROCESS

If X(t) represents the no. of occurrences of certain even in (0, t), then the discrete random process {X(t)} is called the Poisson process, provided the following postulate are satisfied.

 

i.                       P[1 occurrence in (t, t+t) = λ∆t

 

ii.                       P[0 occurrence in (t, t+t) = 1 - λ∆t

 

iii.                       P[2 occurrence in (t, t+t) = 0

X(t) is independent of the no. of occurrences of the event in any interval prior and after the interval (0, t)

v. The Prob. That the events occurs a specified no. of times in (t0, t0+t) depends only on t, but not on t0.

 

 

Prob. Law for the Poisson Process {X(t)}



 

BINOMIAL PROCESS

Let Xn, n = 1, 2, 3, … be a Bernoulli Process and Sn  denote the No. of the successes in the 1st n Bernoulli trails i.e., S


 

Example:3.7.1

Suppose that customers arrive at a bank according to a Poisson Process with mean rate of 3 per minute. Find the Prob. That during a time interval of 2 minutes (i) exactly 4 customer arrive(ii)Greater than 4 customer arrive (iii) Fewer than 4 customer arrive.




Example:3.7.2

 

If customers arrive at a counter in accordance with a Poisson process with a mean rate of 2 per minute, find the Prob. that the interval 6/w two consecutive arrivals is (i) more than 1 minute (ii) B/W 1 & 2 minute (iii) 4 minutes or less

λ= 2


 

TUTORIAL QUESTIONS

 

1.. The t.p.m of a Marko cain with three states 0,1,2 is P and the initial state distribution is Find (i)P[X2=3]  ii)P[X3=2, X2=3, X1=3, X0=2]

2. Three boys A, B, C are throwing a ball each other. A always throws the ball to B and B always throws the ball to C, but C is just as likely to throw the ball to B as to A. S.T. the process is Markovian. Find the transition matrix and classify the states

 

3. A housewife buys 3 kinds of cereals A, B, C. She never buys the same cereal in successive weeks. If she buys cereal A, the next week she buys cereal B. However if she buys P or C the next week she is 3 times as likely to buy A as the other cereal. How often she buys each of the cereals?

4. A man either drives a car or catches a train to go to office each day. He never goes 2 days in a row by train but if he drives one day, then the next day he is just as likely to drive again as he is to travel by train. Now suppose that on the first day of week, the man tossed a fair die and drove to work if a 6 appeared. Find 1) the probability that he takes a train on the 3rd day. 2). The probability that he drives to work in the long run.



 WORKED OUT EXAMPLES



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