# 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,λ_{i}).For every point in time t_{1},X(t_{1}) 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(ω_{o}t)

**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 X_{n}.

– For every n, X_{n }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(t_{1}), X(t_{2}),………, X(t_{n})}

For any set of samples for time {t_{1}, t_{2},………., t_{n}} and for order n.

Denote X_{k}=X(t_{k})

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

fx_{1},…..x_{n}(x_{1},……,x_{n})

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

Px_{1},……,x_{n }(x_{1},……x_{n})

*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

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

- Non stationary

- The probability density function depends on the time origin.
- At least one or more of the mean values will depend on time
- 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(t
_{1})X(t_{2})} depends only on the time difference i.e. t_{1}-t_{2}.

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

- Every number of the random process has the same statistical behavior as the entire random process.
- The statistical behavior can be determined by examining only one sample function.
- Such processes are called ergodic.
- The mean values are determined by time averages.
- Ergodic processes are also stationary processes

- Nonergodic

- It is opposite to ergodic
- 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,λ_{i}).For every point in time t_{1},X(t_{1}) 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.

**GATE Syllabus-**

1.Gate syllabus for Mathematics 2014

2. Gate Syllabus for Physics 2014

3. Gate Syllabus for Electronics and Communication 2014

4.Gate Syllabus for Engineering Science 2014

**IES Syllabus-**

1. IES Syllabus for General ability

2.IES Syllabus for Electronics and Telecomm

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