Probability Theory and Stochastic processing (EC-703 (C)) - Important Questions

State and prove Bayes' theorem of probability.

If A and B are independent events, prove that A' and B, A and B', and A' and B' are also independent.

Given P(A fails)=0.01, P(B fails)=0.03 and P(B fails | A fails)=0.06, find (i) P(both fail), (ii) P(A fails | B fails), (iii) whether A fails and B fails are statistically independent.

For a Gaussian random variable with mean 1 and variance 4, find the probability that the variable lies between 1 and 2.

Verify that the Rayleigh density is a valid probability density function.

Find the moment generating function about the origin of the Poisson distribution.

Find the density of the random variable Z = X + Y, where X and Y are two independent uniform random variables over (-1, 1).

Define the moment generating function and state and prove its properties.

State and prove the properties of the joint distribution function.

Explain briefly about time average and ergodicity.

Check stationarity of the random-amplitude process X(t)=A cos wt with A uniform over (0, pi).

Show X(t)=cos(wt+theta) with theta uniform over (0, pi/2) is not WSS and find its average power.

Prove that the power spectral density (PSD) and the autocorrelation function of a random process form a Fourier transform pair.

Prove that Syy(w)=|H(w)|^2 Sxx(w) for a linear system.

State and prove properties of the cross power spectral density.

Explain the concept and classification of stochastic (random) processes.