1.d.
Define Characteristic function and present generation of moments using it.
(3 marks)
00
1.e.
State central limit theorem for the case of equal distributions.
(2 marks)
00
1.f.
Write the properties of jointly Gaussian random variables.
(3 marks)
00
1.g.
What is a WSS random process?
(2 marks)
00
1.h.
Write short notes on Gaussian random process.
(3 marks)
00
1.i.
Write the expression for power spectral density.
(2 marks)
00
1.j.
Write any three properties of cross-power density spectrum.
(3 marks)
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PART-B
Unit-I
2.
A missile can be accidentally launched if two relays A and B both have failed. The
probabilities of A and B failing are known to be 0.01 and 0.03, respectively. It is
also known that B is more likely to fail (probability 0.06), if A has failed.
What is the probability of an accidental missile launch?
What is the probability that A will fail, if B has failed?
Are the events “A fails” and “B fails” statistically independent?
(10 marks)
00
OR
3.
You (A) and two others (B and C) each toss a fair coin in a two-step gambling
game. In step1 the person whose toss is not a match to either of other two is “odd
man out”. Only the remaining two whose coins match go on to step2 to resolve the
ultimate winner.
What is the probability that you will advance to step2 after the first toss?
What is the probability you will be out after the first toss?
What is the probability that no one will be out after the first toss?
(5 marks)
00
Unit-II
4.a.
Obtain the moment generating function of a uniformly distributed random variable.
(5 marks)
00
4.b.
Obtain the variance of Raleigh random variable.
(5 marks)
00
OR
5.a.
A random variable X uniformly distributed in the interval (0, $\pi$/2). Consider the transformation Y = sinx, obtain the pdf of Y.
(5 marks)
00
5.b.
Obtain the variance of Gaussian random variable.
(5 marks)
00
Unit-III
6.a.
The joint characteristic function of two random variables is given by $\phi XY(\omega_{1},\omega_{2}) = exp(-\omega _1^{2} -4\omega _2^{2})$
(5 marks)
00
6.b.
X and Y are statistically independent random variables and W = X+Y obtain the pdf
of W
(5 marks)
00
OR
7.a.
Write the properties of joint distribution function.
(5 marks)
00
7.b.
Prove that the variance of weighted sum of N random variables equals the weighted sum of all their covariances.
(5 marks)
00
Unit-IV
8.
Define autocorrelation function of a random process. Write properties of auto
correlation function of a WSS process and prove any three of them.
(10 marks)
00
OR
9.a.
A random process X(t) = Acos($\omega_o$t) + B sin($\omega_o$t) where $\omega_o$ is a constant and A, B are uncorrelated zero mean random variables with same variances. Check whether X(t) is WSS or not?
(5 marks)
00
9.b.
Classify random processes and explain
(5 marks)
00
Unit-V
10.
Derive the relationship between cross-power spectrum and cross-correlation
function.
(10 marks)
00
OR
11.a.
The autocorrelation function of a random process $R_{XX}(\tau$) = 4 cos($\omega_o \tau$), where $\omega_o$ is a constant. Obtain its power spectral density.
(5 marks)
00
11.b.
Obtain the average power in the random process X(t) = Acos($\omega_o \tau + \Theta $) where A, $\omega_o$ are real constants and $\Theta $ is a random variable uniformly distributed in the range (0, 2$\pi$).
(5 marks)
00
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