Anna University Chennai 2009-4th Sem B.E Electronics & Communication Engineering ./B.Tech , /E (ester, Electronics and Communication Engineering)-RANDOM PROCESSES - Question Paper
B.EVB.Tech. DEGREE EXAMINATION, MAY/JUNE 2009.
Fourth Semester Electronics and Communication Engineering MA 1254 RANDOM PROCESSES (Regulation 2004)
Time : Three hours Maximum : 100 marks
Answer ALL questions.
PART A (10 x 2 = 20 marks)
In a coin tossing experiment, if the coin shows head, one die is thrown and the result is recorded. But if the coin shows tail, two dice are thrown and their sum is recorded. What is the probability that the recorded number will be 2?
2. Find the value of c given the pdf of a random variable X
as
, if 1 <x < o x
f(x) =
0, otherwise
If the service life in hours of a semiconductor is a random variable having a weibull distribution with parameters a = 0.025 and 0 = 0.5, how long can such a semiconductor be expected to last?
A random variable X has pdf fix) =
>"x, x > 0 0, x < 0
. Find the density function
1
of
X
5. State central limit theorem.
6. If the joint pdf of {X, Y) is given by fix, y) = 2 - x ~ y, in 0<<y<l, find EiX).
7. Prove that the sum of two independent Poisson processes is a Poisson Process.
8. Define sine wave process.
9. Define wide sense stationary process. Give an example.
10. The power spectral density function of a zero mean wide sense stationary
process {X(f)} is given by S((o) = ( M < Find R(r).
[0, elsewhere
PART B (5 x 16 = 80 marks)
11. (a) (i) The chances of A, B and C becoming the general Manager of a
company are In the ratio 4:2:3. The probabilities that the bonus scheme will be introduced in the company if A, B and C become general manager are 0.3, 0.7 and 0.8 respectively. If the bonus scheme has been introduced, what is the probability that A has : been appointed as general manager? (8)
(ii) Out of 2000 families with 4 children each, how many would you expect to have
(1) at least 1 boy
(2) 2 boys
(3) 1 or 2 girls
(4) no girls? (8)
Or
(b) (i) A random variable X has the following distribution x: -2-10 1 2 3 p(x) : 0.1 'k 0.2 2k 0.3 3*
(1) Evaluated
(2) Evaluate P(-2 < X < 3)
(3) Find the cumulative distribution function ofX. (8)
(ii) A continuous Random variable has the pdf f{X) - kxK , - 1 < x < 0.
Find the value of k and P
2 J 3288
12. (a) (i) Define Geometric distribution. Obtain its MGF and hence compute the first four moments. (10)
(ii) For a normal distribution with mean 2 and variance 9, find the value of xx, of the variable such that the probability of the variable lying in the interval (2, rcA) is 0.4115. (6)
Or
(b) (i) A Random variable X has a uniform distribution over the interval (3, 3). Compute
(1) P(X = 2)
(2) P(\X- 2|<2)
(3) Find k such that P{X > k) = 1 / 3. (6)
(ii) Define Gamma distribution. Prove that the sum of independent Gamma variates is a Gamma variate (4)
(iii) In a book of 520 pages, 390 typographical errors occur. Assuming Poissons law for the number of errors per page, find the probability that a random sample of 5 pages will contain no error. (6)
Find the marginal and conditional densities if
13. (a) (i)
f(x,y) = k(x3y + xy3), 0<x <2, 0<y<2. (8)
The joint distribution of (X, Y) where X and Y are discrete is given
(ii)
|
X |
Y | ||
|
0 |
1 |
2 | |
|
0 |
0.1 |
0.04 |
0.06 |
|
1 |
0.2 |
0.08 |
0.12 |
|
2 |
0.2 |
0.08 |
0.12 |
Verify whether X, Y are independent.
(8)
Or
2 xy x +
_ 0<*<1,0<y<2 , 3 elsewhere 'Fmd
0
(b) (i) The joint pdf of a two dimensional random variable (X, Y )is given by fix, y) =
/ I) X>-2
(1)
(2)
P
P(Y < X) /
P
Y <\!x< 2/ 2
(3) (8)
(ii) For 10 observations on price X and supply Y the following data were obtained :
XX = 130, Sy = 220, IX2 =2288, IY2 = 5506 and Z XY = 3467 . Obtain the line of regression of Y on X and estimate the supply when the price is 16 units. (8)
14. (a) (i) Classify the random process and explain with an example. (8)
(ii) Given a random variable Y with characteristic function (p{w) and a random process X(t) = cos{M + Y). Show that {X(0} is stationary in the wide sense if 1) = 0 and <p{2) = 0 . (8)
Or
(b) (i) Three boys Xy Y, Z are throwing a ball to each other. X always throws the ball to Y and Y always throws the ball to Z> but Z is just as likely throw the ball to Y as to X. Show that the process is Markovian. Find the transition probability matrix and classify the states. (8)
(ii) A machine goes out of order whenever a component fails. The failure of this part follows a Poisson process with a mean rate of 1 per week Findthe probability that 2 weeks have elapsed sia.ee last failure. If there arc 5 spare parts of this component in an inventory and that the next supply is not due in 10 weeks, find the probability that the machine will not be out of order in the next 10 weeks. (8)
15. (a) (i) Consider two random processes X(tf) = 3 cos (cot + 0) and
n
Y(t) - 2cos(&Jt + (p) where (p-9and 8 is uniformly
2
distributed random variable over (0,2n). Verify whether jRJfWjQ) (8)
Define Power spectral density and cross spectral density of a random process. State their properties. (8)
(ii)
(b) (i)
Or
If the power spectral density of a wide sense stationary process is
a 1 - a i find the autocorrelation
given by S(co) =
function of the process. (8)
For a linear system with random input X(f), the impulse response hit) and output Y(t), obtain the power spectrum and cross
power spectrum Sio)).
(8)
(ii)
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Attachment: |
| Earning: Approval pending. |
