## Random Signal Analysis - Dec 2014

### Electronics & Telecomm. (Semester 5)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1 (a)** State and explain:

i) Independent Events

ii) Joint and conditional probabilities of events.(4 marks)
**1 (b)** What is moment generating function and why is it called so?(4 marks)
**1 (c)** State central limit theorem with its importance.(4 marks)
**1 (d)** Define Markov chain giving an example.(4 marks)
**1 (e)** Explain Bayes theorem and total probability theorem.(4 marks)
**2 (a)** Two dice with faces 1,2,3,4,5,6 are throw and the sum of the faces is counted. Plot the probability mass function for the sum of the faces. What is the probability that the product of the faces is 12?(10 marks)
**2 (b)** X and Y are two random variables for which joint pdf is given by P(x=i, Y=j)=c(i+j) i=,1,2,3,4; j=1,2,3 find c and conditional mean and variance of X given Y=1.(10 marks)
**3 (a)** For a certain communication channel the probability that '0' received as '0' is 0.8 while the probability that '1' is received as '1' is 0.95. If probability of transmitting 0 is 0.45 find.

i) a '1' is received

ii) '1' was transmitted given that '1' was received

iii) Probability that error has occurred.(10 marks)
**3 (b)** If X and Y are exponential distributions with unity parameter, find the probability distribution function of

U=X+Y V=X(X+Y)(10 marks)
**4 (a)** Let X1, X2, X3,..... be sequence of Random variables. Define:

i) Convergence almost everywhere

ii) Convergence in probability

iii) Convergence in mean square sense

iv) Convergence in distribution.

For the above sequence of the Random variable X.(10 marks)
**4 (b)** What is Power spectral Density? Explain its significance. Find the power spectral density of random process given by X(t)=a cos (bt+Y) where Y is a random variable uniformly distributed over (0,2?).(10 marks)
**5 (a)** Define auto correlation function of a WSS random variable. List the properties of Auto correlation Function of Random Process and prove any two properties. Also give one practical application of Auto correlation function.(10 marks)
**5 (b)** State Chapman-Kolmogorov equation. Transition probability matrix of Markow Chains is $$ \begin{matrix}
\ \ 1 & \ \ \ 2 & \ \ 3
\end{matrix}
\\
\begin{matrix}
1\\2
\\3
\end{matrix}
\begin{bmatrix}
0.5 &0.4 &0.1 \\0.3
&0.4 &0.3 \\0.2
&0.3 &0.5
\end{bmatrix} $$ Find the limiting probabilities.(10 marks)
**6 (a)** Prove that if input LTI system is WSS the output is also WSS. What is Ergodic process?(10 marks)
**6 (b)** A random process is defined by X(t)=10 cos (100t+?) where ? is uniformly distributed in (0, 2?). Verify whether X(t) WSS random Process and correlation ergodic.(10 marks)