Generate KERALA UNIVERSITY Class 4 probability random processes and numerical techniques Question Paper

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401 : PROBABILITY, RANDOM PROCESSES AND NUMERICAL TECHNIQUES (FR) Time : 3 Hours Max. Marks : 100 PART - A Answer all questions. Each question carries 4 marks. :
If $f(x)=\frac{k}{2^{x}}$ is a probability distribution of a random variable which can take values $x=0,1,2,3,4$. Find $K$ and Mean of the distribution.
Find the mean and variance of the probability distribution with density function $f(x)=K e^{-\frac{1}{8}\left(x^{2}+8 x+16\right)}$.
The customers arrive at a bank according to a Poisson Process with mean rate of 2 minutes. Find the probability that during an 1 minute interval no customers arrive.
The autocorrelation function of a stationary process $\{(X(t))\}$ is given by $R(\tau)=2+4 e^{-2|\tau|}$. Find mean and variance of the process $\{(X(t))\}$. P.T.O.