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

State and prove Bayes' theorem of probability.

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.

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

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

State and prove (explain) the properties of the joint distribution function.

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

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

Explain briefly about time average and ergodicity.

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

Show $X(t)=\cos(\omega t+\theta)$ 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 $S_{yy}(\omega)=|H(\omega)|^2 S_{xx}(\omega)$

State and prove properties of the cross power spectral density.