STUPIDSID
1 (a)
State and prove Baye's theorem.
5 M
1 (b)
A certain test for a particular cancer is known to be 95% accurate. A person submits to the test and the result are positive. Suppose that the person comes from a population o 100,000 where 2000 people suffer from that disease. What can we conclude about the probability that the person under test has that particular cancer?
5 M
1 (c)
Let X and Y be independent, uniform r.v.'s in (-1, 1). Compute the pdf of V=(X+Y)2.
5 M
1 (d)
If the spectral density of a WSS process is given by \[ \begin {align*} S(w)&= b (a-|w| )/a &, |w|\le a \\ &=0 &, |w|>a \end{align*} \] Find the autocorrelaton function of the process.
5 M
2 (a)
State and prove Chapman-Kolmogorov equation.
10 M
2 (b)
The joint density function of two continuous r.v.'s X and Y is \[ \begin {align*} f(x,y) & = exy &0< x<4, 1/,y<5 \\ &=0 & otherwise \ \ \ \ \ \ \ \ \ \ \ \ \ \end{align*} \] i) Find the value of constant C.
ii) Find P(X≥3, Y≤2)
iii) Find marginal distribution function of X.
ii) Find P(X≥3, Y≤2)
iii) Find marginal distribution function of X.
10 M
3 (a)
Explain strong law of large number and weak law of large numbers.
5 M
3 (b)
Explain the central limit theorem.
5 M
3 (c)
A distribution with unknown mean μ has variance equal to 1.5. Use central limit theorem to find how large a sample should be taken from the distribution in order that the probability will be at least 0.95 that the sample mean will be within 0.5 of the population mean.
10 M
4 (a)
Given a.r.v. Y with chracteristic function. ϕ(w)=E(ejwT) and a andom process defined by X(t)= cos (λt+Y), show that X(t) is stationary in wide sense if ϕ(1)=ϕ(2)=0.
10 M
4 (b)
Define an ergodic process. Determine whether the stochastic process X(t)=A sin(t)+Bcos(t) is ergodic. Here A & B are normally distributed independent r.v.'s with zero mean and equal standard deviation.
10 M
5 (a)
The joint probability function of two discrete r.v.'s X and Y is given by f(x,y)=e(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(XY), E(X2), E(Y2), var(X), var(Y), cov (X,Y) and ρ.
10 M
5 (b)
State and explain various properties of autocorrelation function and power spectral density function.
10 M
6 (a)
"The transition probability matrix of Markov Chain is \[ \ \ 1 \quad \ \ 2 \ \quad \ 3 \\ \begin{matrix}1\\2 \\3 \end{matrix} \begin{bmatrix} 0.3 &0.4 &0.1 \\0.3 &0.4 &0.3 \\0.2
&0.3 &0.5 \end{bmatrix} \] Find the limiting probabilities."
10 M
Write a short notes on any two of the following:
6 (b) (i)
Markov chains.
5 M
6 (b) (ii)
Little's formula.
5 M
6 (b) (iii
LTI systems with stochastic input.
5 M
6 (b) (iv)
M/G/1 queuing system.
5 M
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