Important RGPV Question
EC-703 (C) Probability Theory and Stochastic processing
VII Sem, EC
Unit I Probability and Random Variable
Q.1 State and prove Bays theorem of probability. (RGPV Nov 2023)
Q.2 If A and B are independent events, prove that the events A’ and B, A and B, and A and B’are also independent. (RGPV Nov 2023)
Q.3 A missile can be accidentally launched if two relays A and B both have failed. The probabilities of A and B failing are known to be 0.01 and 0.03 respectively. It is also known that B is more likely to fail (probability 0.06) if A failed. (RGPV Nov 2023)
i) What is the probability of an accidental missile launch?
ii) What is the probability that A will fail if B has failed?
iii) Are events “A fails” and “B fails” statistically independent.
Q.4 State and prove Bayes’s Theorem. (RGPV Dec 2020)
Q.5 A missile can be accidentally launched if two relays A and B both have failed. The probabilities of A and B failing are known to be 0.01 and 0.03 respectively. It is also known that B is more likely to fail (probability 0.06) if A failed. (RGPV Dec 2020)
i) What is the probability of an accidental missile launch?
ii) What is the probability that A will fail if B has failed?
iii) Are events “A fails” and “B fails” statistically independent?
Q.6 What are the axioms of probability? Give engineering examples. (RGPV Dec 2020)
Unit II Distribution & Density Functions and Operation on One Random Variable
Q.1 A Gaussian random variable has mean value 1 and variance of 4. Find the probability that random variable has value between 1 and 2. (RGPV Nov 2023)
Q.2 Verify that Rayleigh Density is a valid density Function. (RGPV Nov 2023)
Q.3 Find the density of the random variable Z-X+Y, where X and Y are two independent uniform random variables over (-1, 1). (RGPV Nov 2023)
Q.4 Define moment generating function. State and prove the properties of moment generating function. (RGPV Nov 2023)
Q.5 Find the movement generating function of a uniform distribution and hence find its mean obtain the mean. (RGPV Dec 2020)
Q.6 Explain Chebyshev’s Inequality(RGPV Dec 2020).
Q.7 Explain the Rayleigh probability density function. (RGPV Dec 2020)
Unit III Multiple Random Variables and Operations
Q.1 State and prove the properties of Joint distribution function. (RGPV Nov 2023)
Q.2 Obtain mean value of a sum of N weighted random variables. Also define joint moments about the origin. (RGPV Dec 2020)
Q.3 List the properties of N random variable. (RGPV Dec 2020)
Q.4 Explain the properties of joint distribution. (RGPV Dec 2020)
Unit IV Stochastic Processes – Temporal Characteristics
Q.1 Explain briefly about time average and ergodicity. (RGPV Nov 2023)
Q.2 Explain the concept and classification of stochastic process. (RGPV Dec 2020)
Q.3 Explain the following terms: (RGPV Dec 2020)
i) Variance
ii) Skew
Unit V Stochastic Processes – Spectral Characteristics
Q.1 Consider a random process X(t) = cos(cov + theta) where o is a real constant and 8 is a uniform random variable in (0, pi / 2) Show that X(t) not a WSS process. Also find the average power in process. (RGPV Nov 2023)
Q.2 Prove that Power Spectral Density (PSD) and Auto correlation function of Random process form a Fourier transform pair. (RGPV Nov 2023)
Q.3 State and prove any three properties of cross power density spectrum. (RGPV Nov 2023)
Q.4 Derive the relation between cross power density spectrum and cross correlation function of a random process. (RGPV Dec 2020)
Q.5 List the properties of autocorrelation function. (RGPV Dec 2020)
Extra Questions
Q.1 X(t) = is a random process, where ‘A’ is uniform random variable over (0, π). Check X(1) for stationarity. X(1) = ACoswr(RGPV Nov 2023)
Q.2 Prove that S yy ( omega)=|H( omega)|^ 2 S xx ( omega) (RGPV Nov 2023)
Q.3 Classify random processes and explain. (RGPV Nov 2023)
Q.4 In a control system, a random voltage X is known to have a mean value of –2V and a second moment of 9V2. If the voltage X is amplified by an amplifier that gives an output Y = 1.5 X+2, find the variance of Y. (RGPV Dec 2020)
Q.5 Define in-phase component and quadrature-phase component. (RGPV Dec 2020)
Q.6 Write short notes on the following: (RGPV Dec 2020)
i) Relationship between power spectrum and autocorrelation function.
ii) Poisson random process.
— Best of Luck for Exam —
