## Random Signal Analysis - May 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)** If A and B are any two events, then prove that P(A∪B)=P(A)+P(B)-P(A?B)(5 marks)
**1 (b)** Explain the concept of conditional probability with an example(5 marks)
**1 (c) ** State and prove Baye's Theorem and Total Probability Theorem(10 marks)
**2 (a)** Define discrete and continuous random variables, give one example of each type. Define exception of discrete random variable and continuous random variable(10 marks)
**2 (b)** Suppose two million lottery tickets are issued with 100 winning tickets among them.

(i) If a person purchase 100 ticket, what is the probability of winning?

(ii) How many tickets should one buy to be 95% confident of having a winning ticket?(10 marks)
**3 (a)** Find the characteristics function of Poisson distribution and find it's mean and variance(10 marks)
**3 (b)** Let X be a random variable with CDF F_{x}(x) and PDF f(x)(x). Let y=aX+B where a and b are real constant and a ≠0. Fid PDF of y in terms of F_{x}(x)(10 marks)
**4 (a)** Suppose that X and Y are continues random varibles with Joint Probability Density function

$$ f_{xy}(x,y)=\dfrac {xe^{-y}}{2}; \ \ 0<x<2, \ \ y>0 $$ $$=0 \ \ elsewhere $$(10 marks)
**4 (b)** (ii) Find the join cumulative distribution function of X and Y

Find the marginal probability density functions of X and Y.(10 marks)
**5 (a)** Define Central Limit Theorem and give its significance(5 marks)
**5 (b)** Describe sequence of ramdom variables(5 marks)
**5 (c) ** IF two random variable are independent then prove that the density of their sum equals the convolution of their density functions.(10 marks)
**6 (a)** Consider a random process X(t) defined by X(t)=A Cos(ωt+θ); -∞<t<∞ where A and ω are constant and θ is a uniform random variable over (-π, π). Show that X(t) is WSS(10 marks)
**6 (b)** Prove that if the input to a linear time invaiant system is WSS then the output is also WSS(10 marks)
**7 (a)** Explain power spectral density. State its important properties and prove any one property(10 marks)
**7 (b)** State and prove the Chapman-Kolmogorov equation.(10 marks)