## Random Signal Analysis - Dec 2013

### 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)** Define a strict- sense stationary(SSS) and a wide-sense stationary(WSS) random
process.(5 marks)
**1 (b)** Show that the conditional probability satisfies the axioms of probability(5 marks)
**1 (c) ** State and explain Total probability theorem and Bayes theorem.(5 marks)
**1 (d)** State the Central limit theorem and give its significance.(5 marks)
**2 (a)** What is CDF of a random variable? State and prove the properties of Distribution functions.
(10 marks)
**2 (b)** It is known that the screws produced by a certain company will be defective with probability 0.01
independently to each other. The company sells the screws in packages of 10 and offers a money-back guarantee that at most
1 out of 10 screws defective. What proportion of packages sold must the company replace?
(5 marks)
**2 (c) ** The probability of hitting an aircraft is 0.001 for each shot. How many shots should be fired so
that the probability of hitting with two or more shots is above 0.95.
(5 marks)
**3 (a)** Define characteristic function of a Random variable .Prove that the characteristic function of the
n_{th} moment is given by

$$ E[x^n]= \dfrac {1}{j^n}\dfrac {d^n}{dw^n} \ \phi_x \ (w)/w=0 $$(10 marks)
**3 (b)** Suppose pdf of $$ x, \ f_x \ (x)=\dfrac {2x}{\pi^2}, 0<x<\pi $$ and y= sin x, Determine the PDF of Y.
(10 marks)
**4 (a)** Find the normalisation constant C and the marginal pdf?s for the following joint pdf-

$$f_{xy}(x,y)=f(x)=
\left\{\begin{array}{l}ce^{-x}e^{-y},\ \ \ \ 0\leq{}y\leq{}x<\infty{} \\
0,\ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \&elsewhere\end{array}\right.
$$

(10 marks)
**4 (b) ** Explain in brief:-

(i)Poison process

(ii)Gaussian process

```
</span><span class='paper-ques-marks'>(10 marks)</span>
```

**5 (a)** Explain what is a Random process. Define Ensemble mean, auto-correlation and Auto covariance of

the processes in terms of Indexed Random variables in usual Mathematical forms.

```
</span><span class='paper-ques-marks'>(10 marks)</span>
```

**5 (b) ** Consider the Random phase Sinusoid given by X(t)=Acos(wt+ϕ) where A and w are constants and ϕ~u

[0.2π] is a Random variable. Prove that Random phase sinusoid is ergodic in both mean and auto-correlation.

(10 marks)
**6 (a)** Let Z=X+Y. Determine the PDF of Z f_{Z} (z). (10 marks)
**6 (b)** A stationary process is given by x(t)=10cos(100t+θ) where θ is a random variable with uniform

probability distribution in the interval (-π,π) , show that it is wide sense stationary process.

```
</span><span class='paper-ques-marks'>(10 marks)</span>
```

**7 (a)** State and prove the Chapman-Kolmogorov equation. (10 marks)
**7 (b)** The transition probability Matrix of Markov chain is

$$
\left[\begin{array}{
ccc}
0.5 &
0.4 & 0.3 \\
0.3 & 0.4 & 0.3 \\
0.2 & 0.3 & 0.5
\end{array}\right]
$$

Find the limiting probabilities.

```
<br> </span><span class='paper-ques-marks'>(10 marks)</span>
```