Random Signal Analysis - Dec 2012

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 prove Baye's theorem.(5 marks) 1 (b) State the Axiomatic definition of probability(5 marks) 1 (c) If A and B are two events such that :P(A)=0.3,P(B)=0.4,P(A∩B)=0.2.
Find
(i)P(A∪B)

(ii)P((A/B)

(iii)P(A/B)

(iv)P(A∪B)
(5 marks) 1 (d) Explain the properties of distribution function(5 marks) 2 (a) The joint probability distribution of a two dimensional random variable (X,Y) is given by f(x,y) =k x y e^-(x2+y2); x ≥0, y ≥0. Find
(i) The value of k
(ii) Marginal density function of X and Y
iii) Conditional density function of Y given that X=x and Conditional density function of X given that Y=y
Check for independence of X and Y.
(10 marks) 2 (b) Explain moment generating function of discrete random variable and continuous random variable in detail. (10 marks) 3 (a) If X,Y are two independent random variables with identical uniform distribution in(0,1), find the probability density function of (U,V) where U=X+Y and V=X-Y. Are U and V independent. (10 marks) 3 (b) Find the characteristic function of Binomial distribution and Poisson distribution. (10 marks) 4 (a) Define Central limit theorem (5 marks) 4 (b) Describe the sequence of random variables. (5 marks) 4 (c) Explain and prove Chebychev's inequality (10 marks) 5 (a) A random process is given by x(t)=sin(Wt+Y) where Y is uniformly distributed over (0,2π) ,verify whether {x(t)} is a wide sense stationary process (10 marks) 5 (b) State the properties of auto-correlation function and cross-correlation function. (10 marks) 6 (a) If a random process is given by x(t)=10cos(100t+θ) where θ is uniformly distributed over (-π,π) , prove that {x(t)} is correlation ergodic. (10 marks) 6 (b) A WSS random process {X(t)} is applied to the input of an LTI system whose impulse response is te^-at) u(t) where a(>0) is real constants. Find the mean of the output Y(t). (10 marks) 7 (a) State and prove Chapman-Kolmogorov equation. (10 marks) 7 (b) The transition matrix of Markov chain with three states 0,1 and 2 is given by
$$ P=\begin{matrix} 0\\ 1\\ 2 \end{matrix}\begin{bmatrix} 3/4 &1/4 &0 \\ 1/4&1/2 &1/4 \\ 0&3/4 &1/4 \end{bmatrix} $$ and the initial state distribution is $$ P(x_o=i)=1/3, \ i=0,1,2 \\ Find :- \\ (i)\ P[X_2=2]\\ (ii)\ P[X_3=1, \ X_2=2, \ X_1=1, \ X_0=2] $$(10 marks)