Question Paper: Probability Theory and Stochastic Processes Question Paper - Jun 17 - Electronics And Communication Engineering (Semester 3) - Jawaharlal Nehru Technological University (JNTUH)
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## Probability Theory and Stochastic Processes - Jun 17

### Electronics And Communication Engineering (Semester 3)

Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.

PART-A

1.a. Define Random variable
(2 marks) 00

1.b. Write about the continuous and mixed random variables
(3 marks) 00

1.c. Mention the difference between the Variance and Skew.
(2 marks) 00

1.d. Write about the Rayleigh density and distribution function
(3 marks) 00

1.e. Explain the equal and unequal distributions
(2 marks) 00

1.f. Write about linear transformations of Gaussian random variables.
(3 marks) 00

1.g. Mention the properties covariance.
(2 marks) 00

1.h. Show that $S_{xx}(\omega) = S_{xx}(-\omega)$.
(3 marks) 00

1.i. State wiener-Khinchin relation
(2 marks) 00

1.j. Express the relationship between power spectrum and autocorrelation.
(3 marks) 00

PART-B

Unit-I

2.a. Discuss the mutually exclusive events with an example
(5 marks) 00

2.b. Define probability, set and sample spaces.
(5 marks) 00

OR

3. Write the classical and axiomatic definitions of Probability and for a three digit decimal number chosen at random, find the probability that exactly K digits are greater than and equal to 5, for 0<K<3. </div> (10 marks) 00

Unit-II

4.a. Obtain the relationship between probability and probability density function.
(5 marks) 00

4.b. Find the moment generating function of the random variable whose moments are $m_r = (r + 1)!2^r$.
(5 marks) 00

OR

(5 marks) 00

5.b. Determine the moment generating function about origin of the Poisson distribution.
(5 marks) 00

Unit-III

6.a. Differentiate between the marginal distribution functions, conditional distribution functions and densities.
(5 marks) 00

6.b. Given the transformation y= cos x where x be a uniformly distributed random variable in the interval ($−\pi, \pi$). Find $f_y (y)$ and E[y].
(5 marks) 00

OR

7. Let X be a random variable defined, Find E [3X] and E[$X^2$] given the density function as $f_{x}(x)=\begin{array}{cc}{(\pi / 16) \cos (\pi x / 8),} & {-4 \leq x \leq 4} \\ {0,} & {\text { elsewhere }}\end{array}$

(10 marks) 00

Unit-IV

8.a. State and prove properties of cross correlation function.
(5 marks) 00

8.b. If the PSD of X(t) is $S_{xx}(\omega )$. Find the PSD of dx(t)/dt.
(5 marks) 00

OR

9. A random process $Y(t) = X(t)- X(t +\tau )$ is defined in terms of a process X(t). That is at least wide sense stationary.

1. Show that mean value of Y(t) is 0 even if X(t) has a non Zero mean value.

2. If $Y(t) = X(t) +X(t + \tau)$ find E[Y(t)] and $\sigma Y^2$

(10 marks) 00

Unit-V

10. The auto correlation function of a random process X(t) is $R_{XX} (\tau ) = 3+2 exp (−4\tau^2)$.

1. Evaluate the power spectrum and average power of X(t).

2. Calculate the power in the frequency band $-1/ \sqrt{2} \leq \omega \leq 1/ \sqrt{2}$.

(10 marks) 00

OR

11. Derive the relation between PSDs of input and output random process of an LTI system.
(10 marks) 00