Anna University Coimbatore 2011 B.E Electronics

MODEL EXAMINATION

PROBABILITY & RANDOM PROCESSES

Year/Semester & Branch: II / IV / ECE A & B

Max. Marks: 100 Time: 180 min

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

1. The mean of binomial distribution is 20 and standard deviation is 4. Find the parameters of the distribution.
2. Establish the memory less property of Exponential distribution.
3. A continuous random variable X has probability density function . Find the value of A.
4. If X is uniformly distributed in. Find the probability density function of .
5. If X and Y are random variable with joint probability density functionsthen check whether X and Y are independent.
6. State central limit theorem.
7. Write the formula for angle between regression line.
8. The two regression lines are . Find the correlation coefficient.
9. Define wide sense stationary process.
10. What is Markov chain? When a Markov chain is said to be homogeneous?
11. Distinguish between Second order stationary and weakly stationary random process.
12. If the transition probability matrix is , find the limiting distribution of the chain.
13. State any two properties of cross correlation function.
14. If is the auto correlation function of a random process X(t), obtain the spectral density of X(t).
15. Check whether the following system is linear.
16. Define stationary process.
17. Write a note on classification of noise.
18. Define band limited white noise.
19. Check whether the system is causal: i) ii) .
20. Prove that .

 

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

21. A random variable X has the following probability distribution.

x:	-2	-1	0	1	2	3
P(x):	0.1	k	0.2	2k	0.3	3k

 

 

Find i) the value of k, ii) evaluate ,

iii) find the cumulative distribution of X. and iv) evaluate the mean of X.

22. a) Find the Covariance of X and Y if the joint pdf (6 Marks)

b) Ten candidates obtained the following marks in examinations in history and mathematics.

Find the rank correlation between them: (6 Marks)

Candidate	1	2	3	4	5	6	7	8	9	10
History Marks	40	65	61	49	53	42	68	57	58	46
Maths Marks	51	58	67	55	76	45	69	56	73	63

23. a) The process {X (t)} whose probability distribution is given by .

Show that it is not stationary. (8 Marks)

b) Show that the random process is not stationary, if A and are constants

and is uniformly distributed random variable in (4 Marks)

24. a) Consider two random process and where is a

random variable uniformly distributed in prove that . (8 Marks)

b) The auto correlation function of a wide sense stationary random process is given by

.Determine the power spectral density of the process. (4 Marks)

25. a) Consider the random process where B and are independent random variables.

B is a random variable with mean 0 and variance 1. is uniformly distributed in the interval .

Find the mean and Autocorrelation of the process. (8 Marks)

b) Find the power spectral density of a random processif and

(4 Marks)

26. a) The marks obtained by a number of students in a certain subject are assumed to be normally

distributed with mean 65 and standard deviation 5. If 3 students are selected at random from this

group, what is the probability that two of them will have marks over 70? (6 Marks)

b) The joint p.d.f. of two R.Vs X and Y is . Obtain the marginal p.d.f of X and that of Y. Hence or otherwise find. (6 Marks)

27. a) Let be a Markov chain on the space S= {1, 2, 3} with one step transition

probability matrix.And the initial distribution is Find

(8 Marks)

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

mean square value of the process. (4 Marks)

28. a) An engineer analyzing a series of digital signals generated by a testing system observes that only 1 out of 15 highly distorted signals followed a highly distorted signal, with no recognizable signal,

whereas 20 out of 23 recognizable signals follow recognizable signals with no highly distorted signal

between. Given that only highly distorted signals are not recognizable, find the fraction of signals that

are highly distorted. (8 Marks)

b) Find the Autocorrelation of Gaussian White noise. (4 Marks)