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Chapter: Mathematics (maths) - Random Processes

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Important Short Objective Questions and Answers: Random Processes

Mathematics (maths) - Random Processes - Important Short Objective Questions and Answers: Random Processes

Markov Processes and Markov chains

 

 

1. Define Random processes and give an example of a random process. Answer:

 

A Random process is a collection of R.V {X (s,t)}that are functions of a real variable

namely time t where s S and t T

Example: X (t) = Acos(ωt + θ ) where θ is uniformly distributed in ( ,20π ) where A and ω  are constants.

 

2. State the four classifications of Random processes. 

Sol: The Random processes is classified into four types

(i)Discrete random sequence

If both T and S are discrete then Random processes is called a discrete Random sequence.

(ii)Discrete random processes

If T is continuous and S is discrete then Random processes is called a Discrete Random

processes.

(iii)Continuous random sequence

If T is discrete and S is continuous then Random processes is called a Continuous Random sequence.


(iv)Continuous random processes

 

If T &S are continuous then Random processes is called a continuous Random processes.

 

3.       Define stationary Random processes.

 

If certain probability distributions or averages do not depend on t, then the random process

{X (t)}is called stationary.

 

4. Define first order stationary Random processes.

A random processes {X (t)}is said to be a first order SSS process if

 

f (x1 ,t1 + δ ) = f (x1 ,t1 ) (i.e.) the first order density of a stationary process {X (t)} is independent of time t

 

5. Define second order stationary Random processes

 

A RP {X (t)} is said to be second order SSS if f (x1, x2 ,t1 ,t2 ) = f (x1 , x2 ,t1 + h,t2 + h) where f (x1, x2 ,t1,t2 ) is the joint PDF of {X (t1 ), X (t2 )}.

 

6. Define strict sense stationary Random processes

 

Sol: A RP {X (t)} is called a SSS process if the joint distribution X (t1 )X (t21 )X (t3 ).........X (tn ) is the same as that of

 

X (t1 + h)X (t2 + h)X (t3 + h).........X (tn + h) for allt1 ,t2 ,t3 .........tn and h >0 and for n 1.

 

7. Define wide sense stationary Random processes

 

A RP {X (t)}is called WSS if E{X (t)} is constant and E[X (t)X (t +τ )]= Rxx (τ ) (i.e.) ACF is a function of τ only.

8. Define jointly strict sense stationary Random processes

 

Sol: Two real valued Random Processes {X (t)}and {Y(t)} are said to be jointly stationary in the strict sense if the joint distribution of the {X (t)}and {Y(t)} are invariant under translation

 

of time.

 

9. Define jointly wide sense stationary Random processes

Sol: Two real valued Random Processes {X (t)}and {Y(t)} are said to be jointly stationary in the wide sense if each process is individually a WSS process and RXY (t1 ,t2 ) is a function of  t1 ,t2 only.

 

10.      Define Evolutionary Random processes and give an example. 

Sol: A Random processes that is not stationary in any sense is called an Evolutionary process. Example: Poisson process.

 

 

11. If {X (t)} is a WSS with auto correlation R(τ ) = Aeα τ , determine the second order moment of the random variable X (8) – X (5).


E(X 2 (t))= E(X (t)X (t)) = Rxx (t,t) = Aeα (0) = A

 

E(X 2 (8)) = A & E(X 2 (5)) = A E(X (8)X (5)) = Rxx ( 5,)8= A.e3α . 

Now second order moment of {X (8) X (5)} is given by

 

E(X (8) X (5))2 = E(X 2 (8) + X 2 (5) 2X (8)X (5))

 

= E(X 2 (8)) + E(X 2 (5)) 2E(X (8)X (5))

 

= A + A 2Ae3α  = 2A 1( e3α )

 

12.Verify whether the sine wave process{X (t)}, where X (t) = Y cosωt where Y is uniformly distributed in (0,1) is a SSS process.

Sol: F(x) = P(X (t) x) = P(Y cosωtx)


 

If {X(t)} is to be a SSS process, its first order density must be independent of t. Therefore, {X(t)}is not a SSS process.

 

13. Consider a random variable Z(t) = X1 cosω0t X 2 sinω0t where X1 and X2 are independent Gaussian random variables with zero mean and variance σ 2 find E(Z) and E( Z 2 )

Sol: Given E(X1 ) = 0 = E(X 2 ) &Var(X1 ) = σ 2 = Var(X 2 )

E(X12 ) = σ 2 = E(X 2 2 )

E(Z) = E(X1 cosω0t X 2 sinω0t) = 0

E(Z 2 ) = E(X1 cosω0t X 2 sinω0t)2


 

14.Consider the random process X (t) = cos(ω0t + θ ) where θ is uniformly distributed in (π ,π ) .Check whether X(t) is stationary or not?

 

Answer:




15. Define Markov Process.                                                                     

Sol: If for t1         < t2   < t3 < t4 ............ < tn    < t then

P(X (t) x / X (t1 ) = x1 , X (t2 ) = x2 ,..........X (tn ) = xn ) = P(X (t) x / X (tn ) = xn ) Then the process {X (t)}is called a Markov process.

 

16. Define Markov chain.

 

Sol: A Discrete parameter Markov process is called Markov chain.

 

17. Define one step transition probability.

Sol:

 

Yes it is irreducible since each state can be reached from any other state


 

Therefore the chain is irreducible.

 

20 State the postulates of a Poisson process.

 

Let {X (t)} = number of times an event A say, occurred up to time ‘t’ so that the sequence {X (t)}, t 0 forms a Poisson process with parameter λ .

(i)                P[1 occurrence in (t,t + ∆t) ]=λt

 

(ii)              P[0 occurrence in (t,t + ∆t) ]=1-λt

 

(iii)           P[2 or more occurrence in (t,t + ∆t) ]=0

 

(iv)           X(t) is independent of the number of occurrences of the event in any interval prior and after the interval (0,t).

(v)              The probability that the event occurs a specified number of times in (t0,t0+t) depends only on t, but not on t0.

 

 

20.State any two properties of Poisson process Sol: (i) The Poisson process is a Markov process

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

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

 

21.If the customers arrived at a counter in accordance with a Poisson process with a mean rate of 2 per minute, find the probability that the interval between two consecutive arrivals is more than one minute.

Sol: The interval T between 2 consecutive arrivals follows an exponential distribution with


 

 

22.A bank receives on an average λ = 6 bad checks per day, what are the probabilities that it will receive (i) 4 bad checks on any given day (ii) 10 bad checks over any 2 consecutive days.


 

23.Suppose the customers arrive at a bank according to a Poisson process with a mean rate of 3 per minute. Find the probability that during a time interval of 2 minutes exactly 4 customers arrive


 


25.Customers arrive a large store randomly at an average rate of 240 per hour. What is the probability that during a two-minute interval no one will arrive.


 

 

26.The no of arrivals at the reginal computer centre at express service counter between 12 noon and 3 p.m has a Poison distribution with a mean of 1.2 per minute. Find the probability of no arrivals during a given 1-minute interval.


 

27. For the sine wave process X (t) = Y cosωt,−∞ < t < ∞ where ω = constant, the amplitude Y is a random variable with uniform distribution in the interval 0 and 1.check wheather the process is stationary or not.


 

Therefore it is not stationary.

 

28. Derive the Auto Correlation of Poisson Process.

Sol: Rxx (t1 ,t2 ) = E[X (t1 )X (t2 )]

R                        xx (t1 ,t2 ) = E[X (t1 ){X (t2 ) X (t1 ) + X (t1 )}]

=E[X (t1 ){X (t2 ) X (t1 )}]+ E[X 2 (t1 )]

=E[X (t1 )]E[X (t2 ) X (t1 )]+ E[X 2 (t1 )]

 

Since X (t) is a Poisson process, a process of independent increments.

Rxx (t1 ,t2 ) = λt1 (λt2 λt1 ) + λ1t1 + λ12t12 if t2 t1

Rxx (t1 ,t2 ) = λ2t1t2 + λt1 if t2 t1

(or) Rxx (t1 ,t2 ) = λ2t1t2 + λ min{t1 ,t2 }

 

29. Derive the Auto Covariance of Poisson process

Sol: C(t1 ,t2 ) = R(t1 ,t2 ) E[X (t1 )]E[X (t2 )]

=   λ2t1t2 + λt1 λ2t1t2 = λt1 if t2 t1

C(t1,t2 ) = λ min{t1,t2}

 

30.Define Time averages of Random process.

Sol: The time averaged mean of a sample function X (t) of a random process{X (t)} is


mean and variance of X(10)-X(6). 

Answer:

X(10)-X(6) is also a normal R.V with mean µ(10) µ(6) = 0 and Var[X (10) X (6)] = var{X (10)}+ var{X (6)} 2cov{X (10), X (6)}

 

= C(10,10) + C( 6,)62C(10 6,) = 16 +162×16e4 = 31.4139

 

33.Define a Birth process.

 

Answer:

A Markov process {X(t)} with state space S={1,2…..} such that



34.Define Ergodic Random Process.

 

Sol: A random process {X (t)}is said to be Ergodic Random Processif its ensembled averages

are equal to appropriate time averages.

 

35.Define Ergodic state of a Markov chain.

 

Sol: A non null persistent and aperiodic state is called an ergodic state.

 

36.Define Absorbing state of a Markov chain.

 

Sol: A state i is called an absorbing state if and only if Pij = 1 and Pij = 0 for i j

37.Define irreducible

 

The process is stationary as the first and second moments are independent of time. State any four properties of Autocorrelation function.

 

Answer:

i)   RXX (τ ) = RXX (τ )

 

ii)    R(τ ) R(0)

iii)   R(τ ) is continuous for all τ

 

iv) if RXX (τ ) is AGF of a stationary random process {X(t)} with no periodic

components, then µX 2 = lim →∞ R(τ )

38.What do you mean by absorbing Markov chain? Give an example.

 

Sol: A State I of a Markov chain is said to be an absorbing state if Pii = 1(i.e.) it is

 

impossible to leave it .A Markov chain is said to be absorbing if it has at least one absorbing state.

 

Eg: The tpm of an absorbing Markov chain is


 

39.Define Bernoulli Process.

 

Sol: The Bernoulli random variable is defined as X (ti ) = ti which takes two values 0 and 1 with the time index ti such that {X (ti , s) : i = ......... 1 1,,0,......;s = 1,}0

 

40State the properties of Bernoulli Process. Sol: (i) It is a Discrete process

 

(ii) It is a SSS process

 

(iii)   E(Xi ) = p, E(X 2 ) = p and Var(Xi ) = p 1( p)

 

41.Define Binomial Process

Sol: It is defined as a sequence of partial sums {Sn },n = ,13,.....2 where

 

Sn = X1 + X 2 + ........ + X n

 

42.State the Basic assumptions of Binomial process

 

Sol: (i) The time is assumed to be divided into unit intervals. Hence it is a discrete time process.

(ii) At most one arrival can occur in any interval

(iii) Arrivals can occur randomly and independently in each interval with probability p.

 

43.Prove that Binomial process is a Markov process

 

Sol: Sn = X1 + X 2 + ........ + X n Then Sn = Sn1 + X n

 

P(Sn = m / Sn1 = m) = P(X n = 0) = 1 p. Also P(Sn = m / Sn1 = m 1) = P(X n = 1) = p.

 

(i.e.) the probability distribution of Sn ,depends only on Sn1 .The process is Markovian process.

 

44. Define Sine wave process.

Sol: A sine wave process is represented as X (t) = Asin(ωt + θ ) where the amplitude A ,or frequency ω or phase θ or any combination of these three may be random.

It is also represented as X (t) = Acos(ωt + θ ) .


45.Prove that sum of two independent Poisson process is also Poisson.

Sol: Let X (t) = X1 (t) + X 2 (t)


Therefore X (t) = X1 (t) + X 2 (t) is a Poisson process with parameter (λ1 + λ2 )t

 

 

 





 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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