STUPIDSID
Answer the following
1(a)
For an LTI system with stochastic input prove that autocorrelation of output is given by convolution of cross-correlation (between input-output)and LTI system impulse response.
5 M
1(b)
Suppose that a pair of fair dice are tossed and let the RV X denote the sum of the points. Obtain probability mass function and cumulative distribution function for X.
5 M
1(c)
If Z = X+Y and if X and Y are independent then derive pdf of Z as convolution of pdf of X and Y.
5 M
1(d)
Write a note on the Markov chains.
5 M
2(a)
Define and Explain moment generating function in detail.
5 M
2(b)
let Z=X/Y. Determine fz(z)
5 M
2(c)
The joint cdf of a brivariate r.v. (X,Y) is given by \[\begin {align*} F_{XY}(x,y)&=(1-e^{-\alpha x})(1-e^{-\beta y}),x\geq 0, y\geq 0,\alpha,\beta>0\\
&=0\ \text{otherwise}.
\end{align*}\]
i) Find the marginal cdf's of X & Y.
ii) Show that X & Y are independent.
iii) Find P(X≤1, Y≤1), P(X≤1), P(Y>1) & P(X>x, Y>y)
i) Find the marginal cdf's of X & Y.
ii) Show that X & Y are independent.
iii) Find P(X≤1, Y≤1), P(X≤1), P(Y>1) & P(X>x, Y>y)
10 M
3(a)
Explain strong law of large numbers and weak low of large numbers.
5 M
3(b)
Write a note on birth and death queuing models.
5 M
3(c)
A distribution with unknown mean &mu
10 M
4(a)
State and prove Chapman-Kolmogorov equation.
5 M
4(b)
State and prove bayes theorem.
5 M
4(c)(i)
State any three properties of power spectral density.
3 M
4(c)(ii)
If the spectral density of a WSS process is given by \[\begin{matrix}
S(w)=b(a-|w|)/a, & |w|\leq a\\
=0 , & |w|>a
\end{matrix}\]
Find the autocorrelation function of the process.
Find the autocorrelation function of the process.
7 M
5(a)
The joint probability function of two discrete r.v. 's X and Y is given by f(x,y) = c(2x+y), where x and y can assume all integers such that 0≤x≤2, 0≤y≤3 and f(x,y) = 0 otherwise. Find E(X), E(Y), E(X2), E(Y2, var(X), var(Y), cov(X,Y), and ρ.
10 M
5(b)
prove that if input LTI system is WSS the output is also WSS. What is ergodic process?
10 M
6(a)
The transition probability matrix of Markov Chain is \[\begin{matrix}
& \begin{matrix}
1 &2 &3
\end{matrix}\\
\begin{matrix}
1\\
2\\
3
\end{matrix} & \begin{bmatrix}
1 &1 &1 \\
1 &1 &1 \\
1 &1 &1
\end{bmatrix}
\end{matrix}\]
10 M
6(b)
An information source generates symbols at random from a four later alphabet {a, b, c, d} with probabilities P(a)=1/2, p(b)=1/4 and P(c)=P(d)=1/8. A coding scheme encodes these symbols into binary codes as follows:
a 0
b 10
c 110
d 111
Let X be the random variable denoting the length of the code, i.e. the number of binary symbols.
i) What is the range of X?
ii) Sketch the cdf Fx(X) of X, and specify the type of X.
iii) Find P(X≤1, P(11) & P(1≤X≤2).
a 0
b 10
c 110
d 111
Let X be the random variable denoting the length of the code, i.e. the number of binary symbols.
i) What is the range of X?
ii) Sketch the cdf Fx(X) of X, and specify the type of X.
iii) Find P(X≤1, P(1
5 M
6(c)
Write notes on the following
i) Block diagram and explanation of angle & multiple server queuing system
ii) M/M/1/&infin queuing system
i) Block diagram and explanation of angle & multiple server queuing system
ii) M/M/1/&infin queuing system
10 M
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