Anna University Coimbatore 2009 B.E Electronics & Communication Engineering Probability and Random Process-Model - exam paper

MODEL EXAMINATION

PROBABILITY & RANDOM PROCESSES

Year/Semester & Branch: II / IV / ECE

Max. Marks: 100 Time: 180 min

PART-A Answer ALL Questions (20 X 2 = 40)

1. A continuous random variable X has probability density function given by . Find K such that .
2. The number of hardware failures of a computer system in a week of operation has the following probability mass function : No. of failures 0 1 2 3 4 5 6

Probability 0.18 0.28 0.25 0.18 0.06 0.04 0.01

Find the mean of the number of failures in a week.

3. The moment generating function of X is given by . Find .
4. In a normal distribution whose mean is 12 and standard deviation is 2. Find the probability intense from X=9.6 to X=13.8.
5. The joint probability density function of a bivariate random variable (X,Y) is given by . Find .

6.       Find the marginal density function of X and Y if

7. Define Covariance.
8. The two equations of the variables X and Y are and . Find the correlation coefficient between X and Y.
9. Define discrete random process. Give an example.
10. If the transition probability matrix is , find the limiting distribution of the chain.
11. What will be the superposition of independent poisson processes with respective average rates?
12. Give an example of Markov process.
13. State any two properties of an auto correlation function.
14. If , find the mean and variance of the process. (N/D 2006)
15. If is the auto correlation function of a random process X(t), obtain the spectral density of X(t).
16. Given the power spectral density , find the average power of the process.
17. Find the auto correlation function of a Gaussian white noise.
18. Define band limited white noise.
19. Name the commonly used filters in electrical systems and define them mathematically.
20. A white noise signal with power spectral density is applied to an RC low pass filter. Find the auto correlation of the output signal of the filter.

PART-B Answer ANY 5 Questions (5 X 12 = 60)

21. A random variable X has the following probability distribution

. Find (a) the value of k,

(b) Evaluate (c) If find the minimum value of C.

(d) Evaluate (e) Find

22. a) Suppose the joint probability density function is given by . Obtain the marginal probability density of X and that of Y. Hence or otherwise find .

b) Given the joint probability density function of (X,Y) as .

Find the marginal and conditional PDF of X and Y. Are X and Y independent?

23. Given a random variable Y with characteristic function and a random process defined by , show that is stationary in the wide sense of .
24. a) A stationary random process has an auto correlation function and is given by

. Find the mean and variance of the process. (6 Marks)

b) If the auto correlation function of WSS process is , show that its spectral

density is given by . (6 Marks)

25. a) X(t) is the input voltage to a circuit (system) and Y(t) is the output voltage. {X(t)} is

a stationary random process with and . Find , if

the power transfer function is . (6 Marks)

b) If , where A is a constant, is a random variable with a uniform distribution in and {N(t)} is a band limited white noise with a power spectral density . Find the power spectral density of {Y(t)}. Assume that N(t) and are independent. (6 Marks)

26. a) If X is uniformly distributed over (0,10) calculate the probability that

i) X > 6 ii) 3 < X < 8 (5 Marks)

b) Calculate the correlation co-efficient for the following heights (in Inches) (7 Marks)

X:	65	66	67	67	68	69	70	72
Y:	67	68	65	68	72	72	69	71

of fathers X their sons Y

 

27. A man goes to his office by car or catches a train every 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 go by car again as he is to travel by train. Now suppose that on the first day of the week, the man tossed a fair dice and went by car to work if and only if a 6 appeared. Find (i) the probability that he went by train on the third day and (ii) the probability he went by car to work in a long run.
28. a) The power spectral density function of a wide sense stationary process is given by .Find

the auto correlation function of the process. (6 Marks)

b) Given the power spectral density of a continuous process as , find the

mean square value of the process. (6 Marks)