Important 2 marks Questions and Answers of Random Process

'2' Marks Questions and Answers

1. Give an example for a continuous time random process. [A.U. N/D 2004, CBT A/M 2011]

2. Define a Stationary process. [A.U. N/D 2004, CBT M/J 2010]

Solution:

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

i.e., the random processes X(t₁) and X(t₂) where t₂ = t₁ + ∆ will have all statistical properties the same.

3. State the four types of a stochastic processes. [A.U. A/M 2004] [A.U Trichy A/M 2010, CBT M/J 2010, A/M 2019 R13 PQT, RP]

Solution:

(i) Discrete time, discrete state random process.

(ii) Discrete time, continuous state random process.

(iii) Continuous time, discrete state random process.

(iv) Continuous time, continuous state random process.

4. Prove that a first order stationary random process has a constant mean. [A.U. Model] [A.U A/M 2011, N/D 2013]

Solution:

See Page No. 3.7 Theorem 1.

5.Define (a) wide sense stationary random process, (b) ergodic random process. (or) When a random process said to be Ergodic process? [A.U Trichy M/J 2011, M/J 2012, A/M 2017 (R8, R13)] [A.U. Model] [A.U N/D 2017 (RP) R13][A.U A/M 2019 R8 RP] [M/J 2013]

Solution:

(a) Wide sense stationary random process :

A random process X(t) is said to be wide sense stationary if its mean is constant and its autocorrelation depends only on time difference.

i.e., (i) E (X (t)) = constant. (ii) R_XX (t,t+τ) = R_XX (τ)

(b) Ergodic Random Process : [A.U CBT A/M 2011, CBT N/D 2011] [A.U M/J 2013]

A random process X (t) is called ergodic if its ensemble averages are equal to appropriate time averages.

6. Give an example of an ergodic process. [A.U. A/M 2004]

Solution :

1. A Markov chain finite state space.

2. A stochastic process X(t) is. ergodic if its time average tends to the ensemble average as T → ∞

7. What is a Markov chain? When can you say that a Markov Chain is homogeneous ? [A.U. N/D 2004, N/D 2010] [A.U N/D 2015, R-13]

Solution

If, for all n,

 is called a Markov Chain (a₁, a₂, ....a_n, are called the states of the Markov Chain.

If P_ij (n-1, n) = P_ij (m-1, m), the Markov chain is called a homogeneous Markov chain.

8. Consider a Markov chain with 2 states and transition probability matrix P =  Find the steady state probabilities of the chain. [A.U. Model]

Solution:

Let the stationar, probabilities  of the chain be π = (π₁ π₂)

By the property of π, π P = π

Since the above equations are identical, consider this equation with π₁ + π₂ = 1

π₁ = 2π₂

.'. 3π₂ = 1 => π₂ = 1/3 and π₁ = 2/3

Hence the steady-state probabilities of the chain is π = (π₁ π₂)

i.e., π = (2/3 1/3)

9. Define irreducible Markov chain? And state Chapman Kolmogorow theorem. [A.U. N/D 2003]

Solution :

If eeeeee for some n and for all i & j, then every state can be reached from every other state. When this condition is satisfied, the Markov chain is said to be irreducible. The tpm of an irreducible chain is an irreducible matrix.

Chapman-Kolmogorov theorem : [A.U A/M 2017 R-13]

If 'P' is the tpm of a homogenous Markov chain, then the n-step tpm P⁽ⁿ⁾ is equal to Pⁿ

10. Define transition probability matrix. [tpm] [AU N/D 2011]

Solution :

The transition probability matrix (tpm) of the process {x_n, n ≥ 0} is defined by

where the transition probabilities (elements of P) satisfy P_ij ≥ 0,

11. What is meant by steady-state distribution of Markov chain? [A.U. A/M 2003] [A.U N/D 2017 R-08]

Solution:

If a homogeneous Markov chain is regular, then every sequence of state probability distribution approaches a unique fixed probability distribution called the stationary (state) distribution or steady-state distribution of the Markov chain.

 where the state probability distribution at step n,  and the stationary distribution at π = (π₁, π₂, ... π_k) are row vectors.

12. Consider a Markov Chain with state {0, 1} and transition probability matrix eeeeee. Is the state 0 periodic? If so what is the period ? [A.U. A/M 2006]

Solution :

Starting from 0→1→0, after 2 transitions it comes back to state 0.

Therefore, period of state 0 is 2.

In the same way period of state 1 is 2.

13. What will be the superposition of 'n' independent Poisson processes with respective average rates λ₁, λ₂, ... λ_n? [A.U. N/D 2004]

Solution:

The superposition of 'n' independent Poisson processes with average rates λ₁, λ₂, … λ_n is another Poisson process with average rate λ₁ + λ₂ + ... + λ_n

14. State any two properties of a Poisson process. [A.U. A/M 2003] [A.U CBT A/M 2011, A.U N/D 2015 R-13 RP, A.U N/D 2017 R-13] [A.U. A/M 2018 R-13] [A.U N/D 2018 R13 PQT, RP]

The Poisson process is a Markov process:

(i) The Poisson process possess the Markov property.

(ii) Sum of two independent Poisson processes is a Poisson process.

(iii) Difference of two independent Poisson processes is not a Poisson process.

15. State the postulates of a Poisson process. [A.U. N/D 2003, A/M 2011, N/D 2017 (RP) R13, N/D 2019 R17 PQT]

Solution:

Let X(t) = number of times an event A say, occured upto time 't' so that the sequence {X(t), t ε [0, ∞)} forms a Poisson process with parameter 'λ'.

(i) Events occuring in non-overlapping intervals are independent of each other.

(ii) P[X(t) = 1 for t in (x, x + h)] = λh + 0(h)

(iii) P[X(t) = 0 for t in (x, x + h)] = 1 - λh + 0(h)

(iv) P[X(t) = 2 or more for t in (x, x + h)] = 0(h)

17. What do you man by an absorbing Markov chain? Give an example.

Solution:

A state i of a Markov chain is said to be an absorbing state if P_ii = 1, i.e., if it is impossible to leave it. A Markov chain is said to be absorbing if it has atleast one absorbing state.

18. Write down the relation satisfied by the steady-state distribution and the tpm of a regular Markov chain.

Solution:

If π = (π₁ π₂ ... π_n) is the steady-state distribution of the chain whose tpm is the nth order square matrix P, then πP = π

19. When is a Markov chain completely specified?

Solution:

A Markov chain is completely specified when the initial probability distribution and the tpm are given.

20. What is a Stochastic matrix? When is it said to be regular?

Solution:

In a square matrix the sum of all the elements of each row is 1, is called a Stochastic matrix. A stochastic matrix P is said to be regular if all the clements of pⁿ (for some positive integer n) are positive.

21. Check whether the following matrix is stochastic? A =  Is it a regular matrix?

Solution :

Definition:

Regular matrix : See Page No. 3.90

I^st row sum = 0 + 1 =1

II^nd row sum = 1/2 + 1/2 = 1

.'.The given matrix is a stochastic.

All the entries of A⁽²⁾ are positive.

.'. Given matrix A is regular.

22. Is a Poisson process a continuous time Markov Chain? Justify your answer. [A.U A/M. 2010]

Solution:

Yes.

By the property of Poisson process, the conditional probability distribution of X(t₃) given all the past values X(t₁) = n₁, X(t₂) = n₂ depends only on the most recent value X(t₂) = n₂.

i.e., The Poisson process possesses the Markov property.

Poisson process with Markov which take discrete values, whether t is discrete or continuous are called Markov Chain.

Hence, Poisson process is a continuous time Markov Chain.

23. What is meant by one-step transition probability ?[A.U Trichy N/D 2010][A.U N/D 2017 R-13]

Solution:

The conditional probability P[X_n = a_j/X_n-1 = a_i] is called the one-step transition probability from state 'a_i' to state 'a_j' at the n^th step and is denoted by P_ij (n-1, n)

24. Is Poisson process stationary? Justify. [A.U A/M 2015 (R-13, R-8)][A.U N/D 2017 R-08]

Solution :

Poisson process is not a stationary process, as its statistical properties (mean, autocorrelation, ...) are time dependent.

25. What is a random process? When do you say a random process is a random variable? [A.U A/M 2015 (R-13, R-8)]

Solution:

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

It is fixed, then the random process is a random variable.

26. Consider a Markov chain with state {0, 1, 2} and transition probability matrix  Draw the state transition diagram. [A.U N/D 2015 R-8]

Solution :

Given:

27. Give an example of evolutionary random process. [A.U A/M 2015 (R-13, R-8)] [A.U N/D 2018 R13 RP]

Solution :

Poisson process.

28. What is Markov process? [AU N/D 2009] [A.U Tvli A/M 2009] [A.U M/J 2016 R13 RP, N/D 2019 R17 RP, A/M 2019 R17 RP] [A.U N/D 2019 R17 PQT]

Solution:

A random process X(t) is said to be a Markov process if for any t₁ < t₂ < t₃ < ... < t_n

(i.e.,) the conditional distribution of X(t) for given values of X(t₁), X(t₂)... X(t_n-1) depends only on X(t_n-1)

29. Let X (t) be a wide-sense stationary random process with slo E[X(t)] = 0 and Y(t) = X(t) - X(t + τ), τ > 0. Compute E[Y(t)] and Var[Y(t)] [A.U N/D 2019 R17 RP]

Solution:

Given: Y(t) = X(t) - X(t + τ)