Download Goa University BA LLB-5 Years 2017 April 2nd Semester General English II Question Paper

VASAVI COLLEGE OF ENGINEERING (Autonomous), HYDERABAD

B.E. (ECE) VI Semester

Probability Theory and Stochastic Processes (Open Elective-II)

Time: 3 hours Max. Marks: 60

Note: Answer all questions from Part A and Part B.

Part A (10 x 2 = 20 Marks)

1. Define conditional probability.

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2. State Baye's theorem.
3. Define moment generating function.
4. Define marginal distribution function.
5. Define stochastic process.
6. Define stationary process.

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7. Define autocorrelation function.
8. Define ergodicity.
9. Write any two properties of power spectral density.
10. Define white noise.

Part B (4 x 10 = 40 Marks)

1. a) In a binary communication system, a ‘1’ or ‘0’ is transmitted. Because of noise in the system, a transmitted ‘1’ is sometimes received as a ‘0’, and a transmitted ‘0’ is sometimes received as a ‘1’. Let p denote the probability that a transmitted ‘1’ is received as a ‘1’, and let q denote the probability that a transmitted ‘0’ is received as a ‘0’. Suppose that the probability that a ‘1’ is transmitted is a.
   

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   i) Find the probability that a ‘1’ is received.
   ii) Find the probability that a ‘1’ was transmitted given that a ‘1’ was received.
   OR
   b) Define probability density function. Write the properties of Gaussian random variable.
2. a) Two random variables X and Y have joint density
   

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   f_XY(x,y) = K(x+y), & 0 < x < 1, 0 < y < 1 \\ 0, & Otherwise
   Find
   i) The value of K
   ii) Marginal densities of X and Y
   OR
   

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   b) The joint density function of two random variables X and Y is given by
   f_XY(x,y) = e^-(x+y), x = 0, y = 0
   Find the density function of the random variable Z = X/Y.
3. a) Define Poisson random process and derive the probability distribution function of Poisson random process.
   OR
   

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   b) A random process is given by X(t) = A cos(?t + ?), where A and ? are constants and ? is a uniform random variable over (0, 2p). Show that X(t) is a wide sense stationary process.
4. a) Find the power spectral density of the response of a linear time-invariant system whose input is a WSS process.
   OR
   b) Derive the relation between the power spectral density and autocorrelation function.

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