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I Semester M.E. (E&C) Degree Examination, March 2013 (2K9 Scheme) EL114 : STOCHASTIC PROCESSES Time : 3 Hours

Max. Marks : 100

Instruction : Answer any five full questions. 1. a) Let A1, A2, A3,.,.,.,.,.,.,An be partitions of sample space Ω and B is an arbitrary event. Show that P(A i B ) =

P (B A i ) P(A i ) P (B A 1) P(A 1) + P (B A 2 ) P( A 2 ) +

















P(B A n ) P(A n )

.

b) The probability that a driver will have an accident in 1 month equals 0.02. Find the probability that in 100 months he will have three accidents. c) Find mean and variance of a random variable X which has uniform distribution in the interval [b, a] using MGF. d) A fair coin is tossed 3 times and random variable X equals number of heads in each trial. Find the pdf and cdf of random variable X. (5+5+5+5) 2. a) Let X and Y be two random variables. Show that P (x1 < X < x 2, Y ≤ y) = F (x2, y) – F (x 1, y). b) Two RVs X and Y have a joint pdf ⎧k x 2y fX, Y (x, y) = ⎨ ⎩ 0

0
Find : i) the constant k so that fX, Y(x, y) indeed is valid joint pdf, ii) find the marginal pdfs of X and Y and, iii) are X and Y independent. c) Let Z = X + Y – c, where X and Y are independent random variables with variances σ 2X and σ 2Y and c is a constant. Find the variance of Z in terms of σ 2X , σ 2Y and c.

(4+10+6) P.T.O.

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3. a) Two independent Random variables X1 and X2 have Poisson distribution with parameters α 1 and α 2 respectively. Find the density function of random variable Y = X1 + X2. b) State convergence in probability and convergence in mean square sense. Show that convergence in mean square sense implies convergence in probability. ⎡2 ⎤ ⎢ 1⎥ c) Suppose X is a 4-variate Gaussian distribution with mean vector, μ X = ⎢ ⎥ ⎢ 1⎥ ⎢ ⎥ ⎣0 ⎦

and covariance matrix

∑

⎡6 ⎢3 ⎢ X= ⎢2 ⎢ ⎣1

3

2

4

3

3

4

2

3

1⎤ 2⎥ ⎥ and X = [X , X , X , X ]T. 1 2 3 4 3⎥ ⎥ 3⎦

i) Can you suggest simple modification for the covariance matrix ∑ X so that the random variables X1, X2, X3, X4 are independent, hence write the expression for the joint pdf of random variables X1, X2, X3, X4. ⎡X ⎤ ii) If Y = ⎢ 1 ⎥, then find the pdf random vector Y. ⎣X2 ⎦

(6+8+6)

4. a) State central limit theorem. The resistors R1, R2, R3, R4 are independent RVs and each is uniform in the interval [450, 550]. Using the central limit theorem, find P {1900 ≤ R1 + R 2 + R 3 + R 4 ≤ 2100 } .

b) A random process consists of three sample functions X (t, s1) = 2, X (t, s 2) = 2 cos (t), and X (t, s3) = 3 sin (t), each occurring with equal probability. Find the mean and autocovariance of the process.

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c) Suppose that X (t) is a random process with autocorrelation function R XX ( τ) = 25 +

4 , find the 1+ 6τ2

i) mean, ii) variance, and iii) covariance of, random random variables Z = X (6) and W = X (9).

(8+6+6)

5. a) For a real valued stationary random process X (t) show that its power spectral ∞

density is even, and that S XX ( f ) = ∫ R XX (t ) cos(2 πft )dt . 0 b) Consider a random process defined by X(t) = A cos(ωt + φ), where A and ω

are constants and φ is a random variable uniform in the interval [− π, π ] . Check whether the process is mean ergodic and correlation ergodic.

c) Suppose that X (t) is normal random process with mean η(t) = 3 and covariance C (t1, t2) = 4 exp (–0.2|t 1 – t2|). Determine i) P [X (5) ≤ 5] and ii) P [|X(8) – X (5) | ≤ 1]. (6+8+6) 6. a) Let X (t) and Y (t) be two zero mean WSS processes and let Z (t) = X (t) + Y (t). Show that, RZZ(0) = RXX (0) + RYY (0), if X (t) and Y (t) are orthogonal and jointly WSS. b) Let X (t) be a WSS random process which is applied as input to a LTI system with impulse response h (t). Derive the expression for the autocorrelation of the output of the system, in terms of its impulse response and autocorrelation of the input process. Hence show that the power spectral density of the output process, Syy(f) can be expressed as, SYY (f) = S XX (f) |X(f)|2. c) Let X1, X2, ....... , Xn, be ‘n’ independent random variables with same mean μ and variance σ 2 . Show that the sample mean of these RV’s is an unbiased and consistent estimator of μ . Find the MSE of the estimator. (4+10+6)

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7. a) Assume that the weather in a certain location can be modeled as a homogeneous Markov chain whose transition probability matrix is given below : Tomorrow's weather Today's weather

Fair

Cloudy

Rain

Fair

0.8

0.15

0.05

Cloudy

0.5

0.3

0.2

Rain

0.6

0.3

0.1

i) Draw the state transition diagram, and ii) If PT (0) = [0.4, 0.2, 0.4], find P (1), P (2), and P (4). b) For the second order autoregressive process, show that Φ 2, 1 = rxx(1)

2 1 − rxx(2) rxx (2) − rxx (1) , Φ = 2, 2 and . 2 2 1 − rxx(1) 1 − rxx (1)

c) Discuss any two properties of matched filters.

(8+6+6)

8. a) Suppose that the under hypothesis H1 the output voltage source is constant m and under H0 it is zero. Before observation, the voltage is corrupted by an 2 additive Gaussian noise of zero mean and variance σ which is independent of the source output. The composite signal x (t) is sampled obtaining a total of N independent samples. Show that the minimum probability of error criterion yields the test. N

∑ i=1

x i2

> <

2

⎛ p ⎞ Nm ln⎜⎜ ⎟⎟ + m ⎝q⎠ 2

H1 σ H0

where xi are samples of x (t), and p and q are the a priori probabilities of H1 and H0 respectively. b) Write short notes on : i) Maximum likelihood estimator and ii) ARMA process. __________________

(8+6+6)