2. (a) What is gaussian random variable? Develop an equation for guassian distribution

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1 Code No: R Set No. 1 II B.Tech I Semester Supplementary Examinations, February 2007 PROBABILITY THEORY AND STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering, Electronics & Telematics and Electronics & Computer Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks 1. (a) Discuss Joint and conditional probability. (b) When are two events said to be mutually exclusive? Explain with an example? (c) Determine the probability of the card being either red or a king when one card is drawn from a regular deck of 52 cards. [6+6+4] 2. (a) What is gaussian random variable? Develop an equation for guassian distribution function. (b) verify that the following is a distrbution function. F(x) = 0 forx < a, = 1 / 2 (x/a + 1)for a, <= x <= a, and = 1forx > a..[10+6] 3. (a) State and prove properties of variance of a random variable (b) Let X be{ a random variable defined by the density function π cos(πx/8) 4 x 4 f X (x) = 16 Find E[3X] and E[X 0 elsewhere 2 ]. [8+8] 4. (a) State and prove central limit theorem. (b) Find the density of W = X + Y where the densities of X and Y are assumed to be f X (x) == [u(x) u(x 1)], f Y (y) = [u(y) u(y 1)], [8+8] 5. (a) Show that the variance of a weighted sum of uncorrected random variables equals the weighted sum of the variances of the random variables. (b) Two random variables X and Y have joint characteristic function φx, Y(ω 1,ω 2 ) = exp( 2ω 2 1 8ω 2 2). i. Show that X and Y are zero mean random variables. ii. are X and Y are correlated. [8+8] 6. (a) Aircraft arrive at an airport according to a poisson process at a rate of 12 per hour. All aircrafts are handled by one air traffic controller. If the controller takes a 2 minute coffee break, what is the probability that he will miss one or more arriving aircrafts. (b) Telephone calls are initiated through an exchange at the average rate of 75 per minute and are described by a poisson process. Find the probability that more than 3 calls are initiated in any 5 second period. (c) A random process is described by X(t) = A. where A is a continuous random variable uniformly distributed on (0,1) classify the process. [6+6+4] 1 of 2

2 Code No: R Set No (a) A WSS noise process N(t) has ACF R NN (τ) = Pe 3 τ. Find PSD and plot both ACF and PSD (b) Find R Y Y (τ) and hence S Y Y (ω ) interns of S XX (ω ) for the product device shown in figure 1 if X(t)is WSS. [8+8] Figure 1: 8. (a) Define the following random processes i. Band Pass ii. Band limited iii. Narrow band. [3 2 = 6] (b) A Random process X(t) is applied to a network with impulse response h(t) = u(t) exp (-bt) where b > 0 is ω constant. The Cross correlation of X(t) with the output Y (t) is known to have the same form: R XY (τ) = u(τ)τ exp (-by) i. Find the Auto correlation of Y(t) ii. What is the average power in Y(t). [6+4] 2 of 2

3 Code No: R Set No. 2 II B.Tech I Semester Supplementary Examinations, February 2007 PROBABILITY THEORY AND STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering, Electronics & Telematics and Electronics & Computer Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks 1. (a) Is probability relative frequency of occurrence of some event? Explain with an example. (b) Define an event and explain discrete and continuous event with an example. (c) One card is drawn from a regular deck of 52 cards. What is the probability of the card being either red or a king? [6+6+4] 2. (a) Define density function. Derive an equation for binomial distribution function (b) What is the distribution function of mixed random variable? Discuss what do you mean by density function. [8+8] 3. (a) In an experiment when two dice are thrown simultaneously, find expected value of the sum of number of points on them. { 1 (b) The exponential density function given by.f X (x) = b e (x a)/b x > a 0 x < a Find out variance and coefficient of skewness. [6+10] 4. Discrete random variables X and Y have a joint distribution function F XY (x, y) = 0.1u(x + 4)u(y 1) u(x + 3)u(y + 5) u(x + 1)u(y 3)+ 0.05u(x)u(y 1) u(x 2)u(y + 2) u(x 3)u(y 4)+ 0.12u(x 4)u(y + 3) Find (a) Sketch F XY (x, y) (b) marginal distribution functions of X and Y. (c) P( 1 < X 4, 3 < Y 3) and (d) Find P(X < 1, Y 2). [ ] 5. (a) For two zero mean Gaussian random variables X and Y show that their joint characteristic function is φxy(ω 1,ω 2 ) = exp{ 1/2[σX 2 ω 2 1+2ρσ X σ Y ω 1 ω 2 +σy 2 ω 2 2]}. (b) Statistically independent random variables X and Y have moments m 10 = 2, m 20 = 14, m 02 = 12 and m 11 = -6 find the moment µ 22. (c) Two Gaussian random variables X and Y have variances σx 2 = 9 and σy 2 = 4, respectively and correlation coefficient ρ. It is known that a coordinate rotation by an angle Π/8 results in new random variables Y 1 and Y 2 that are uncorrelated. what is ρ. [8+4+4] 1 of 3

4 Code No: R Set No Let X(t) be a stationary continuous random process that is differentiable. Denote its time derivative by Ẋ(t). [ ] (a) Show that E (t) = 0. (b) Find R (τ) in terms of R (τ)sss [8+8] 7. (a) A WSS noise process N(t) has ACF R NN (τ) = Pe 3 τ. Find PSD and plot both ACF and PSD (b) Find R Y Y (τ) and hence S Y Y (ω ) interns of S XX (ω ) for the product device shown in figure 1 if X(t)is WSS. [8+8] Figure 1: 8. (a) Determine which of the following impulse response do not correspond to a system that is stable or realizable or both and state why i. h(t) = u(t+3) ii. h(t) = u(t) e t2 iii. h(t) = e + sin(ω 0 t), ω 0 : real constant. iv. h(t) = u(t)e 3t, ω 0 : real constant (b) A random process X(t) = A Sin (ω 0 t +θ ) where A & ω0 are real positive constants & θ is a random variable uniformly distributed in the internal (-π,π ) is applied to the network shown in figure 2. Find an expression for the networks response? [8+8] Figure 2: 2 of 3

5 Code No: R Set No. 2 3 of 3

6 Code No: R Set No. 3 II B.Tech I Semester Supplementary Examinations, February 2007 PROBABILITY THEORY AND STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering, Electronics & Telematics and Electronics & Computer Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks 1. (a) show that two elements cannot be both mutually exclusive and statistically independent. What is the condition for two events to be independent? (b) One card is selected from an ordinary 52-card deck with an event A as select a king, B as select a jack or queen and c as select a heart. Determine whether A,B and C are independent by pairs. (c) What is the probability of drawing 3 white and 4 green balls from a bag that contains 5 white and 6 green balls, if 7 balls are drawn simultaneously at random? [6+6+4] 2. (a) What is gaussian random variable? Develop an equation for guassian distribution function. (b) verify that the following is a distrbution function. F(x) = 0 forx < a, = 1 / 2 (x/a + 1)for a, <= x <= a, and = 1forx > a..[10+6] 3. (a) In an experiment when two dice are thrown simultaneously, find expected value of the sum of number of points on them. { 1 (b) The exponential density function given by.f X (x) = b e (x a)/b x > a 0 x < a Find out variance and coefficient of skewness. [6+10] 4. (a) Define conditional distribution and density function of two random variables X and Y (b) The joint probability density function of two random variables X and Y is given by { a(2x + y f(x, y) = 2 ) 0 x 2, 2 y 4. Find 0 elsewhere i. value of a ii. P(X 1, Y > 3). [8+8] 5. (a) Consider a probability space S = (Ω, F, P).LetΩ= {ξ1... ξ5} = { 1, 1/2, 0, 1 / 2, 1} with P{ξ i } = 1/5 i = Define two random variables on S as follows. X(ξ) =ξandy(ξ) =ξ 2 i. Show that X and Y are dependent random variables. ii. Show that X and Y are uncorrelated. 1 of 2

7 Code No: R Set No. 3 (b) Let X and Y be independent random variables each N(0,1). Find the mean and variance of Z = X 2 +Y 2. [8+8] 6. (a) Determine if the constant process X(t) = A where A is a random variable with mean? A and variance σa 2 is mean ergodic. (b) let N(t) be a zero mean wide sense stationary noise process for which R NN (τ) = (N o /2) δ(t) where N o > 0 is a finite constant. Determine whether N(t) is mean ergodic. (c) A random process consists of three sample functions x(t, s 1 ) =2, x(t,s 2 ) =2 cos t and x(t,s 3 ) =3 sin t each occurring with equal probability. Is the process stationary in any sense. [4+6+6] 7. (a) The PSD of random process is given by { π, ω < δ XX (ω) = 0, elsewhere Find its Auto correlation function. (b) State and Prove any four properties of PSD. [8+8] 8. A random noise X(t) having power spectrum S XX (ω) = 3 49+ω 2 is applied to a to a network for which h 2 (t) = u(t)t 2 exp( 7t). The network response is denoted by Y(t) (a) What is the average power is X(t) (b) Find the power spectrum of Y(t) (c) Find average power of Y(t). [5+6+5] 2 of 2

8 Code No: R Set No. 4 II B.Tech I Semester Supplementary Examinations, February 2007 PROBABILITY THEORY AND STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering, Electronics & Telematics and Electronics & Computer Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks 1. (a) Discuss the following in brief: i. Permutations ii. Combinations iii. Probability (b) How many permutations are there for four cards taken from a 52 card deck? [12+ 4] 2. (a) Define cumulative probability distribution function. And discuss distribution function specific properties. (b) The random variable X has the discrete variable in the set { 1, 0.5, 0.7, 1.5, 3} the corresponding probabilities are assumed to be {0.1, 0.2, 0.1, 0.4, 0.2}. plot its distribution function and state is it a discrete or continuous ditribution function. [8+8] 3. (a) State and prove properties of variance of a random variable (b) Let X be{ a random variable defined by the density function π cos(πx/8) 4 x 4 f X (x) = 16 Find E[3X] and E[X 0 elsewhere 2 ]. [8+8] 4. Discrete random variables X and Y have a joint distribution function F XY (x, y) = 0.1u(x + 4)u(y 1) u(x + 3)u(y + 5) u(x + 1)u(y 3)+ 0.05u(x)u(y 1) u(x 2)u(y + 2) u(x 3)u(y 4)+ 0.12u(x 4)u(y + 3) Find (a) Sketch F XY (x, y) (b) marginal distribution functions of X and Y. (c) P( 1 < X 4, 3 < Y 3) and (d) Find P(X < 1, Y 2). [ ] 5. (a) Show that the variance of a weighted sum of uncorrected random variables equals the weighted sum of the variances of the random variables. (b) Two random variables X and Y have joint characteristic function φx, Y(ω 1,ω 2 ) = exp( 2ω 2 1 8ω 2 2). i. Show that X and Y are zero mean random variables. 1 of 2

9 Code No: R Set No. 4 ii. are X and Y are correlated. [8+8] 6. X = 6 and RXX (t, t+τ) = exp ( τ ) for a random process X(t). Indicate which of the following statements are true based on what is known with certainty X(t) (a) is first order stationary (b) has total average power of 61W (c) is ergodic (d) is wide sense stationary (e) has a periodic component (f) has an ac power of 36w. [16] 7. (a) Consider a random process X(t) = cos (ωt + θ ) where ω is a real constant and θ is a uniform random variable in (0, π /2)). Show that X(t) is not a WSS process. Also find the average power in the process (b) State and prove wiener-khinchin relation. [8+8] 8. (a) Derive the expression for effective input noise temperature of cascaded system in terms of their individual input noise temperatures. (b) Find the available power gain of the following circuit 1 [6+10] Figure 1: 2 of 2

for valid PSD. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I

for valid PSD. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I Code: 15A04304 R15 B.Tech II Year I Semester (R15) Regular Examinations November/December 016 PROBABILITY THEY & STOCHASTIC PROCESSES (Electronics and Communication Engineering) Time: 3 hours Max. Marks: