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## 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)