Random process

Jun 20 • General • 11837 Views • 1 Comment on Random process

Random Process

Introduction

A random process is also known as stochastic process.

A random process X(t) is used to explain the mapping of an experiment which is random with a sample space S which contribute to sample functions X(t,λi).For every point in time t1,X(t1) is a random variable.

  • t represents time and it can be discrete or continuous.
  • The range of t can be finite, but generally it is infinite. It means the process contains infinite number of random variables.

Example: Rolling A Dice

Let random Variable is X=j, where j is the value displayed on top of the dice, after rolling.

Random process is Y(t)=Xcos(ωot)

Ways Of Viewing A Random Process

Lets take a process X(t,ω).

  • When t is fixed, X(t,ω) is a random variable and is known as a time sample.
  • When ω is fixed, X(t,ω) is a deterministic  function of t and is known as realization or a sample path or sample function.

Where ω brings randomness in X(t,ω). In further notations, ω is implied implicitly so it is generally suppressed.

  • When t belongs to countable set, the process is discrete-time.

n -> it denotes the time index

Random process can be written as X(n,ω) or Xn.

–         For every n, Xn is random variable, which can be discrete, continuous or mixed.

  • When t belongs to uncountable infinite set, the process is continuous-time.

Random process can be written as X(t).

Specifying A Random Process

  • A random process can be specified completely by collecting the joint cumulative distribution function among the random variables

{ X(t1), X(t2),………, X(tn)}

For any set of samples for time {t1, t2,………., tn} and for order n.

Denote Xk=X(tk)

–         If process is continuous then it can be expressed by collection of joint probability density function

fx1,…..xn(x1,……,xn)

–         If process is discrete then it can be expressed by collection of joint probability mass  function.

Px1,……,xn (x1,……xn)

Deterministic  And Non-Deterministic Random Process

When the future values of any sample function are predicted depending on the knowledge of the past values, then the random process is known as deterministic random process.

A random process is the combination of time functions, the value of which at any given time cannot be pre-determined. So it is known as non-deterministic process.

Example:-

Lets take a random process {X(t)=A.cos(ωt+θ): t  ≥  0}.

This process has a family of sine waves and depends on random variables A and θ. So it is a deterministic random process.

Stationary And Non Stationary Processes

  • Stationary
  1. All joint density functions of the random process do not depend on the time origin.
  2. Here the mean values are fixed and it does not depend on the time with absolute values.

 

  • Non stationary
  1. The probability density function depends on the time origin.
  2. At least one or more of the mean values will depend on time
  3. We generally take stationary random variables, but this assumption may not be accurate in real situations, but considered in approximate one.

If a random process satisfies the following conditions:

  • The mean of X(t) does not depend on time t, i.e. X(t)=X.
  • The correlation between any two r.v.s E{X(t1)X(t2)} depends only on the time difference i.e. t1-t2.

Then it is called a stationary process in the wide sense.

Stationary in wide sense

  • Stationarity in wide sense is a special case of second-order stationarity.
  • Second order stationarity requires

For every  and. If  ,then the above equation becomes

 Ergodic and Nonergodic Random Processes

  • Ergodic
  1. Every number of the random process has the same statistical behavior as the entire random process.
  2. The statistical behavior can be determined by examining only one sample function.
  3. Such processes are called ergodic.
  4. The mean values are determined by time averages.
  5. Ergodic processes are also stationary processes
  • Nonergodic
  1. It is opposite to ergodic
  2. There is a possibility that stationary processes can be  non ergodic.

IMPORTANT QUESTION AND ANSWERS

1.       What is a random process?

Ans: A random process is also known as stochastic process.A random process X(t) is used to explain the mapping of an experiment which is random with a sample space S which contribute to sample functions X(t,λi).For every point in time t1,X(t1) is a random variable.

2.What is a non-deterministic process?

Ans:A random process is the combination of time functions, the value of which at any given time cannot be pre-determined. So it is known as non-deterministic process.

3.       What is a stationary process?

Ans: In stationary process the joint density functions of the random process do not depend on the time origin.

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One Response to Random process

  1. semere hiluf says:
    at

    propability and random process

    Reply

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