Practice Problem Set - 2

Random Processes

EC-501 (Modelling and Simulation)Spring 2014

1. Let Xn be a sequence of IID Bernoulli random variables with p = 1/2. Find the joint PMF of any ktime samples.

2. Let Dn = 2In − 1, where In is the Bernoulli random process. Find the mean and variance of Dn.

3. Let Xn be a Gaussian random process with zero mean and variance σ2. Find the joint PDF of thecorresponding sum process at times n1 and n2.

4. Inquries arrive at a recorded message device according to a Poisson process of rate 15 inquiries perminute. Find the probability that in a 1 minute period, 3 inquiries arrive during the first 10 secondsand 2 inquiries arrive during the last 15 seconds.

5. Consider a random process X(t) that assumes the values ±1. Suppose that X(0) = ±1 with probability1/2, and that X(t) changes polarity with each occurance of an event in a Poisson process of rate α.Find the PMF of X(t), and hence, its mean and variance.

6. Let X(t) = A for all t, where A is a zero-mean, unit-variance random variable. Find the limiting valueof the time average.

7. A discrete-time random process Xn is defined as follows. A fair coin is tossed. If the outcome is heads,Xn = (−1)n for all n; if the outcome is tails, Xn = (−1)(n+1) for all n. Find the PMF of Xn and thejoint PMF of Xn and Xn+k. Find the mean of Xn.

8. Let Sn be a Binomial counting process. Find the probability that Sn1 = j conditioned on the fact thatSn2

= i, where n2 > n1.

9. Customers arrive at a soft drink dispensing machine according to a Poisson process with rate λ. Supposethat each time a customer deposits money, the machine dispenses a soft drink with probability p. Findthe PMF for the number of soft drinks dispensed in time t.

10. Noise impulses occur on a telephone line according to a Poisson process of rate λ. Suppose that themessage is encoded so that errors caused by a single impulse can be correted. What is the probabilitythat a t-second message is either error-free or correctable?

11. Messages arrive at a computer from two telephone lines according to independent Poisson processes ofrates λ1 and λ2, respectively.

(a) Find the probability that a message arrives first on line 1.

(b) Find the PDF for the time until a message arrives on either line.

(c) Find the PMF for N(t), the total number of messages that arrive in an interval of length t.

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(d) Generalize the result of part (c) for the merging of k-independent Poisson processes of ratesλ1, λ2, . . . , λk, respectively.

N(t) =

k∑i=1

Nk(t)

12. Suppose that the time required to service a customer in a queueing system is an exponential randomvariable with parameter β. If customers arrive at the system according to a Poisson process withparameter λ, find the PMF for the number of customers that arrive during one customer’s servicetime.

13. Let Z(t) be the random signal obtained by switching between the values 0 and 1 according to theevents in a counting process N(t), where

Pr{N(t) = k} =1

1 + λt

(λt

1 + λt

)k

, k = 0, 1, 2, . . .

Find the PMF of Z(t) and its mean value as a function of time.

14. Prove that the moving average process Yn = (1/2)(Xn + Xn−1), where Xn is a Bernoulli randomvariable with p = 1/2, is not a Markov process.

15. A Markov model for packet speech assumes that if the nth packet contains silence, then the probabilityof silence in the next packet is 1 − α. Similarly, if the nth packet contains speech activity, then theprobability of speech activity in the next packet is 1 − β. Find the steady state probabilities for thisprocess.

16. Find the steady state probabilities for a Poisson process with parameter α.

17. Let Mn denote the sequence of random means from an IID random process Xn:

Mn =1

n

n∑i=1

Xi

Is Mn a Markov process? If yes, find the state transition PDF, given by fMn(x |Mn−1 = y).

18. An urn initially contains five black balls and five white balls. The following experiment is repeatedindefinitely : A ball is drawn from the urn; if the ball is white, it is put back in the urn, oterwise itis left out. Let Xn be the number of black balls remaining in the urn after n draws from the urn. IsXn a Markov process? If so, find the appropriate state transition probabilities. Do these probabilitiesdepend on n?

19. Let Xn be the Bernoulli IID process and let Zn be a vector process, defined by Zn = (Xn, Xn−1).Show that Zn is a Markov process and draw its state transition diagram, with the appropriate statetransition probabilities.

20. A critical part of a machine has an exponentially distributed lifetime with parameter λ. Suppose thatn spare parts are initially in stock. Let N(t) be the number of spare parts left at time t. Find thestate transition probability matrix.

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