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

**1.b.**Write about the continuous and mixed random variables

**1.c.**Mention the difference between the Variance and Skew.

**1.d.**Write about the Rayleigh density and distribution function

**1.e.**Explain the equal and unequal distributions

**1.f.**Write about linear transformations of Gaussian random variables.

**1.g.**Mention the properties covariance.

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

**1.i.**State wiener-Khinchin relation

**1.j.**Express the relationship between power spectrum and autocorrelation.

**PART-B**

**Unit-I**

**2.a.**Discuss the mutually exclusive events with an example

**2.b.**Define probability, set and sample spaces.

**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.

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

**OR**

**5.a.**Write about Chebychev’s inequality and mention about its characteristic function.

**5.b.**Determine the moment generating function about origin of the Poisson distribution.

**Unit-III**

**6.a.**Differentiate between the marginal distribution functions, conditional distribution functions and densities.

**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].

**OR**

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

**Unit-IV**

**8.a.**State and prove properties of cross correlation function.

**8.b.**If the PSD of

**is $S_{xx}(\omega )$. Find the PSD of**

*X(t)***.**

*dx(t)/dt***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.

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

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

**Unit-V**

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

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

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

**OR**

**11.**Derive the relation between PSDs of input and output random process of an LTI system.