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

Transcription

1 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: 70 PART A (Compulsory Question) 1 Answer the following: (10 X 0 = 0 Marks) (a) What is the condition for a function to be a random variable? (b) Define Gaussian random variable. (c) How interval conditioning is different from point conditioning? (d) When N random variables are said to be jointly Gaussian? (e) Explain about strict-sense stationery processes. (f) Where the Poisson random processes is used? Explain. (g) (h) (i) (j) ω Examine the function for valid PSD. ω 6 + 3ω + 3 Correlate CPSD and CCF. Analyze the power density spectrum of response. List the properties of band limited processes. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I (a) Give Classical and Axiomatic definitions of Probability. (b) In a single through of two dice, what is the probability of obtaining a sum of at least 10? 3 (a) What is the concept of Random Variable? Explain with a suitable example. (b) A random variable X has the distribution function: F ( x) = X 1 n= 1 n u( x n) 650 Find the probabilities (i) P{- < X 6.5}. (ii) P{X > 4} (iii) P{6 < X 9}. 4 (a) State and explain the central limit theorem. (b) Given the function: UNIT II fxy(x, y) = b(x + y), - < x <, - 3 < y 3. 0, elsewhere (i) Find a constant b such that this is a valid density function. (ii) Determine the marginal density functions f x (x) and f y (y). 5 (a) What are the properties of Jointly Gaussian Random variables? (b) A random variable X has X 3, X = 11, andσ =. For a new random variable Y= X-3, find: (i) (ii) (iii) = x Contd. in page Page 1 of

2 Code: 15A04304 R15 UNIT III 6 (a) List and explain various properties of Autocorrelation function. (b) Given the Autocorrelation function of the processes: RXX( τ ) = τ Find the mean and variance of the process X(t). 7 (a) Compare the Cross Correlation Function with Autocorrelation function. (b) Assume that an Ergodic random process X(t) has an autocorrelation function: RXX( τ) = τ [1 + 4 cos(1τ)] (i) Find X. (ii) Does this process have periodic component? (iii)what is the average power in X(t)? UNIT IV 8 (a) State and explain the Wiener-Khintchine relation. (b) Obtain the auto correlation function corresponding to the power density spectrum: SXX(ω ) = 8 (9 + ω ) 9 (a) Define Power Spectral Density? List out its properties. (b) 6ω Compute the average power of the process having power spectral density ω UNIT V 10 (a) What is LTI system? How the response can be obtained from LTI system. (b) Find the system response, when a signal x(t)= u(t) e -t is applied to a network having an impulse response h(t) = 3u(t) e -3t. 11 (a) Explain about mean and mean square value of system response? (b) A random process X(t) is applied to a network with impulse response: h(t) = u(t) t e -3t. The cross correlation of X(t) with the output Y(t) is known to have the same form R XX (τ) = u(τ) τ e -3τ. (i) Find the autocorrelation of Y(t). (ii)what is the average power in Y(t) Page of

3 Code: 13A04304 R13 B.Tech II Year I Semester (R13) Regular & Supplementary Examinations December 015 PROBABILITY THEY & STOCHASTIC PROCESSES (Electronics and Communication Engineering) Time: 3 hours Max. Marks: 70 PART A (Compulsory Question) 1 Answer the following: (10 X 0 = 0 Marks) (a) What are the conditions to be satisfied for the statistical independence of three events A, B and C? (b) Show that. (c) Two random variables X and Y have the following values: (d) (e) (f) (g) Find the correlation coefficient. Define the joint moments about the origin. Define WSS random process. Determine the mean-square value of a random process with autocorrelation function: A random process has the power density spectrum Find the average power in the process. (h) Define rms bandwidth of the power spectrum. (i) Impulse response of a linear system is The input to this system is a sample function from a random process having an autocorrelation function of Find the autocorrelation of the output. (j) A stationary random process with a mean of is passed through an LTI system with Determine the mean of the output process. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I (a) Define conditional distribution and density functions and list their properties. (b) A continuous random variable X has a PDF Find a and b such that: (i). (ii). 3 (a) Define random variable and give the concept of random variable with an example. (b) The probability density function of a random variable has the form, where is the unit step function. Find the probability that UNIT II 4 (a) Define marginal density and distribution functions. (b) Let X and Y be jointly continuous random variables with joint probability density function: Find: (i) (ii). (iii) Are X and Y independent? 5 (a) State central limit theorem for the following two cases: (i) Equal distributions. (ii) Unequal distributions. (b) Let and zero elsewhere. Find. Contd. in page Page 1 of

4 Code: 13A04304 R13 UNIT III 6 (a) A random process has sample functions of the form: Where A is a random variable uniformly distributed from 0 to 10. Find the autocorrelation function of this process. (b) Show that 7 (a) Show that the autocorrelation function of a stationary random process is an even function of (b) Give the classification of random processes. UNIT IV 8 (a) A stationary random process has a two-sided spectral density given by: Find the mean-square value of the process if (b) List the properties of power spectral density function. 9 (a) For two jointly stationary random processes, the cross-correlation function is. Find the two cross-spectral density function. (b) List the properties of cross power spectral density function. 10 Show that 11 Write short notes on the following: (a) Bandpass random process. (b) Band-limited random process. UNIT V Page of

5 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 013/14 PROBABILITY THEY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE) Time: 3 hours Max Marks: 70 1 (a) Define the following with example: (i) Sample space. (ii) Event. (iii) Mutually exclusive event. (iv) Independent event. (v) Exhaustive event. (b) When a die is tossed find the probabilities of the event A = {odd number shown up}; B = {Number larger than 3 shown up} then find out A B and A B. (a) Define probability density function and explain with an example and write its properties. (b) The random variable has following density function: (i) Find value of k. (ii) P ( k k k 3k 3 (a) The density function of a random variable X is: 1 (b) (i) E [x], (ii) E [(x-1) ], (iii) E [3x -1]. For Poisson distributions find out moment generating function and characteristic function. 4 (a) Distinguish between joint distribution and marginal distribution. (b) Joint probability density function of two random variables X and Y. Find: (i) Value of a ; (ii). 5 (a) Explain relation between marginal and joint characteristic function. (b) In a control system a random voltage is known to have mean value and a second moment If the voltage x is amplified by an amplifier that gives an output, and 6 (a) State the conditions for wide sense stationary random process. (b) Explain the classifications of random process. 7 (a) What is an ergodic random process, present the necessary expression to support the argument? (b) Consider a random process where is a random variable uniformly distributed over where is any real number find 8 (a) Explain the relation between power spectrum and auto correlation function of random process. (b) Write any two properties of cross power density spectrum.

6 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 013/14 PROBABILITY THEY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE) Time: 3 hours Max Marks: 70 1 (a) State and prove Baye s theorem of probability. (b) Two similar boxes A and B contain white and 3 red balls 4 white and 5 red balls respectively. If a ball is selected at random from one of the boxes, then find the probability that the box is B when the ball is red. (a) Write the method for defining conditional event. (b) Check whether following is a probability density function or not 3 (a) Explain variance and skew. (b) The mean and variance of binomial distribution are 4 and 4/3 respectively find (c) Find the expected value of the number on a die when thrown. 4 (a) Define and explain the conditional properties. (b) The joint probability density function of two random variables x and y given by (i) Find the value of c. (ii) Marginal distribution function X and Y. 5 (a) Write the expression for expected value of a function of random variables and prove that the mean value of a weighted sum of random variables equals the weighted sum of mean values. (b) Two Gaussian random variables X 1 and X defined by the mean and co-variance matrices two new random variable s Y 1 and Y are formed using the transformation. Find the matrices, and also find the torrelation co-efficient of and i.e.,. 6 (a) Explain the types of statistical averages. (b) Define a random process where A is Gaussian random variable with zero mean and variance is X(t) is stationary in any sense. 7 (a) Define cross correlation and write its properties. (b) A random process is defined as where is a uniform random variable over (0, Verify the process is ergodic in the mean sense and auto correlation sense. 8 (a) Write the relation between cross power spectrum and cross correlation function. (b) If power spectral density of a random process is given by Find the auto correlation function.

7 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 013/14 PROBABILITY THEY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE) Time: 3 hours Max Marks: 70 1 (a) What is sample space? Explain the discrete and continuous sample space. (b) In a box there are 100 resistors having resistances and tolerances as shown in the table. Resistance Tolerance 5% 10% Total Total Let a resistor be selected from the box and assume each resistor are equally likely occur. Define three events A as draw a 47 Ω resistor, B as draw a resistor with 5% tolerance and C as draw a 100 Ω resistor. (i) The probability of drawing a 47 Ω resistor given that the resistor drawn is 5%. (ii) The probability of drawing a 47 Ω resistor given that the resistor drawn is 100 Ω. (iii) The probability of drawing a resistor of 5% tolerance given resistor is 100 Ω. (iv) Find remaining conditional probabilities. (a) Sketch probability density function and probability distribution function of: (i) Exponential distribution. (ii) Rayleigh distribution. (iii) Uniform distribution. (b) Define conditional distribution function and write their properties. 3 (a) State and prove any three properties of variance of a random variable. (b) For binomial density prove: (i). (ii). (iii) = npq. 4 (a) Define and explain joint distribution function and joint density function of two random variables X and Y. (b) The joint space for two random variable X and Y and its corresponding probabilities are shown in table 1, 1, 3, 3 4, Find and plot (a). Marginal distribution of X and Y. (c) (d). Contd. in Page 3 Page 1 of

8 3 5 (a) Show that the variance of a weighted sum of uncorrelated random variables equals the weighted sum of the variance of the random variables. (b) Two random variables X and Y have the density function: (i) Find all the order moment. (ii) Find covariance. (iii) X and Y and uncorrelated. 6 (a) Explain the concept of random process. (b) Distinguish between: (i) Deterministic random process and non-deterministic random process. (ii) Stationary and non- stationary random process. 7 (a) S.T Where A is normal random variable with zero mean and unity variance and uniformly distributed is Assume A and are independent random variable. (b) Discuss Gaussian random process and state its properties. 8 (a) State and explain four properties of power density spectrum of a random process. (b) If find the spectral density function, where a and b are constant. Page of

9 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 013/14 PROBABILITY THEY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE) Time: 3 hours Max Marks: 70 1 (a) Give the classical and axiomatic definitions of probability. (b) A card is drawn from a well shuffled pack of playing cards. What is the probability that in either a spade or ace? (c) What is the probability: (i) Leap year selected at random will contain 53 Sundays? (ii) Non leap year selected at random will contain 53 Sundays? (a) Define random variable and explain types of random variable. (b) In an experiment of rolling a die and flipping a coin the random variable ( is chosen such that. (i) A coin tail (H) outcome corresponds to positive value of that are equal in magnitude to twice the number that shown on die. Map the elements of random variable into points on the real line and explain. 3 (a) Write about transformation. (b) Let x be a random variable defined by density function: 4 Find. 4 (a) Explain method of finding the distribution and density function for a sum of statistically independent random variables. (b) Find constant b: Is a valid joint density function. 5 (a) Two Gaussian random variable and have variance = 4 respectively and correlation coefficient rotation by an angle results in new random variable Y 1 and Y are uncorrelated what is (b) Prove the mean value of weighted sum of random variables equal to the weighted sum of mean sum of mean value. 6 (a) A random process is defined by where A is a continuous random variable uniformly distribution on (0, 1) determine if it is wide sense stationary. (b) Explain the classification of random process with neat sketch. 7 (a) Write and explain the properties of auto correlation wide sense stationary random process. (b) If the random process X(t) had no periodic components and if X(t) is non-zero mean then: Lim 8 (a) For a random process x(t) derive expression for power density spectrum. (b) A random noise X(t) having power spectrum is applied to a network for which h(t) = The network response is denoted by Y(t): (i) Find the average power of X (t). (ii) Find the power spectrum of Y(t). (iii) Find the average power of Y (t).

10 1 B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 01 PROBABILITY THEY & STOCHASTIC PROCESSES (Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering) Time: 3 hours Max. Marks: (a) Explain the following: (i) Random experiment (ii) Trial (iii) Event (iv) sample space. (b) Find the probability of obtaining 14 with 3 dice using Baye s theorem.. (a) Explain with an example discrete, continuous and mixed random variables. (b) Explain CDF with its properties. 3. What is a characteristic function? Explain its properties with its proofs. 4. (a) State and explain Central Limit Theorem. (b) The joint pdf of two random variable X and Y is given by f XY (x, y) = K, (x + y); x = 0,1,, y = 1,,3,4 0 otherwise Find (i) The K value (ii) P(X = 1, Y = ) (iii) P(X 1, Y 3). (iv) f X (x)& f Y (y) (v) f Y X y 1 & f X Y x. 5. (a) Explain about joint moments about the origin with an example. (b) X is a random variable with mean 4 and variance 3. Another random variable Y is related to X as Y=X+7. Determine (i) E[X ] (ii) E[Y] (iii) var [Y] (iv) R XY 6. (a) Differentiate WSS & SSS. (b) Prove the following: (i) R XX (τ) R XX (0).(ii) R XX ( τ) = R XX (τ). (iii) R XX (0) = E[X (t)]. 7. (a) What is meant by co-variance and explain its properties. (b) A random process X(t) = Acosω 0 t + Bsinω 0 t, where ω 0 is constant and A & B are random variables. If A and B are uncorrelated zero mean having same variance σ but different density functions then show that X(t) is a wide sense stationary. 8. (a) Give the relation between cross power spectrum and cross correlation function. (b) A random process has a power spectrum S XX (ω) = 4 ω, 9 ω 6 0 Find (i). Average power (ii) RMS bandwidth. elsewhere

11 B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 01 PROBABILITY THEY & STOCHASTIC PROCESSES (Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering) Time: 3 hours Max. Marks: (a) Explain the following: (i) Principles of counting (ii) probability as a relative frequency. (b) How many positive integers less than 1000 have no common factor with 1000?. (a) Define a random variables and give the conditions for a function to be a random variable. (b) Explain about normal distribution with its properties. 3. Find the mean and variance of Binomial Distribution and Poisson Distribution. 4. (a) Explain about joint density function with its properties. (b) The joint density function of two random variables X ad Y is given by f XY (x, y) = 1 π 3 e 3 (x xy +y ). Determine the marginal probability density function f X (x) and f Y (y). 5. (a) Explain about jointly Gaussian Random variables. (b) A random variable Z has pdf f Z (z) = ae a(z b) u(z b). Show that the characteristic function of z is z (ω) = a a jw e jwb has the probability function P(x) = 1 3 x x = 1,,3 N., 6. (a) What is random process and classify it and explain. (b) A stationary continuous random process X is differentiable and X (t) is its derivates. Show that E X (t) = 0 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) If X(t) is WSS, find R YY (τ) and hence S YY (ω) in terms of S XX (ω) for the product device shown in below fig. x(t) y(t) S XX (ω) product S YY (ω) Acos ωt 8. If a random process X(t) = A 0 (cos ω 0 t + θ), where A 0 & ω 0 are constants and θ is a uniformly distributed random variable in the interval (0, π) Find (i) Whether X(t) is WSS process? (ii) Power in X(t) by time averaging of its second moment. (iii) The power spectral density ofx(t).

12 3 B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 01 PROBABILITY THEY & STOCHASTIC PROCESSES (Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering) Time: 3 hours Max. Marks: (a) Explain the following: (i) probability (ii) Axioms. (b) From the urn containing n balls any numbers of balls are drawn. Show that the probability of drawing an even number of ball is (n 1 1) ( n 1). (a) Differentiate pmf and pdf. (b) Show that f X (x) = 1 σ π ) e(x μ σ, α 0 < xx < α 0, σ > 0 is a distribution function. 3. (a) The pdf of the random variable X follows f X (x) = 1 θ e x θ θ, α 0 < xx < α 0 Find m.g.f. Hence or otherwise find E(X) and var(x). 4. Joint pdf f XY (x, y) of two continuous random variables X and Y is given by f XY (x, y) = K, e (x+y) for x, y 0; 0 otherwise Where K is constant. (i) find K value and f X (x) and f Y (y). (ii) Are X and Y are statistically independent? (iii) Determine joint CDF and marginal distribution function. (iv) Determine the conditional density functions. 5. (a) Explain about joint central moments with an example. (b) A random variable Z with pdf f Z (z) = 1 ; 1 Z 1. Another random variable random variables (RV)s X = Z and Y = Z. Show that X and Y are uncorrelated. 6. (a) What is meant by stochastic process and classify with an example to each. (b) Check the following for WSS. (i) R XX (t, t + τ) = cosst e t+τ ) (ii) R XX (t, t + τ) = sinτ/(1 + τ ) (iii) R XX (t, t + τ) = 10 τ (iv) R X (t, t + τ) = 5e τ 7. (a) Explain in detail the cross power spectral density. (b) If X(t) and Y(t) are random processes. Prove that (i). S XY (ω) = S YX ( ω) = S YX (ω) (ii) S XY (ω) = S YX (ω) if X(t) & Y(t) are uncorrelated WSS random processes. 8. (a) Give the relation between power spectrum and auto correlation function. (b) Find the cross correlation function corresponding to the cross power spectrum S XY = 6/[(9 + ω )(3 + jω) ]

13 4 B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 01 PROBABILITY THEY & STOCHASTIC PROCESSES (Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering) Time: 3 hours Max. Marks: (a) State the Baye s theorem and prove it. (b) There are 300 students in a class room. It is known that 180 can program JAVA, 10 in C++ 30 in SQL, 1 in JAVA and SQL, 18 in C++ and SQL, 1 in JAVA and C++ and 6 in all three languages. (i) A student is selected at random. What is the probability that she can program in exactly two languages. (ii) Two students are selected at random. What is the probability that they can (a) Both program in JAVA (b) Both program only in JAVA.. (a) Explain about pdf. (b) The diameter of a cable X is taken to be a random variable with pdf f X (x) = 6x(1 x), 0 x 1 (i) verify f X (x) is a pdf or not.(ii) Determine b such that P(x < bb) = P(x > bb). 3. (a) Define and explain moments of a random variable. (b) Find the moment generating function of a random variable. 4. (a) Find the density of W=X+Y, where the densities of X and Y to be f X (x) = u(x) u(x 1)& f Y (y) = u(y) u(y 1). (b) Explain the following (i) Joint distribution function. (ii) conditional distribution function (iii) Marginal distribution function. 5. (a) Show that the variance of variance of a weighted sum of uncorrelated random variables equals the weighted sum of t he variances of the random variables. (b) Explain about transformation of multiple random variables. 6. Explain about the concept stationarity in detail connected with stochastic processes. 7. (a) Explain about WSS and prove any two properties of it. (b) Y(t) = X(t) cos(ω 0 t + θ), where X(t) is a random process and θ is uniformly distributed over the interval (0,π). Determine under what conditions is Y(t) wide seuce stationary. Assume θ and X(t) are statistically independent and ω 0 is constant. 8. (a) Write different types of band pass processes with band limited processes. (b) Find the rms band width of the power spectrum Acos πω S XX (ω) = W, ω W 0, ω W Where ω > 0 & WW > 0 are constants?

14 Code: 13A04304 R13 B.Tech II Year I Semester (R13) Regular Examinations December 014 PROBABILITY THEY & STOCHASTIC PROCESSES (Electronics and Communication Engineering) Time: 3 hours Max. Marks: 70 PART A (Compulsory Question) 1 Answer the following: (10 X 0 = 0 Marks) (a) State Baye s theorem. (b) Three coins are tossed in succession. Find out the probabilities of occurrence of two consecutive heads. (c) State central limit theorem. (d) Find the expected value of the face value while rolling fair die? (e) Define cross-covariance function. (f) Give any two examples for poisson random process. (g) A random process has the power density spectrum S XX (ω) =. Find the average power in the (h) (i) (j) process. What is power spectral density? Mention its importance. Define the following random process: (i) Band limited. (ii) Narrow band. What are the two conditions that are to be satisfied by the power spectrum to be a valid power density spectrum? PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT - I (a) A pack contains 4 white and green pencils, another contains 3 white and 5 green pencils. If one pencil is drawn from each pack, find the probability that (i) Both are white. (ii) One is white and another is green (b) Explain about joint and conditional probability. 3 (a) Consider the experiment of tossing four fair coins. The random variable X is associated with the number of tails showing. Compute and sketch the CDF of X. (b) Define probability density function. List its properties. UNIT - II 4 (a) Let X and Y be jointly continuous random variables with joint density function f XY (x,y) = xy ; for x>0, y>0 = 0; otherwise (i) Check whether x and y are independent. (ii) Find P (x 1, y 1). (b) How expectation is calculated for two random variables? 5 (a) Prove the following: Var (ax+by) = a var(x) + b var(y) + ab cov(x,y) (b) Explain central limit theorem. Contd. in page Page 1 of

15 Code: 13A04304 R13 UNIT - III 6 (a) Explain about mean-ergodic process. (b) If x (t) is a stationary random process having mean = 3 and auto correlation function: R XX (τ) = 9 +. Find the mean and variance of the random variable. 7 (a) Explain the significance of auto correlation. (b) Find auto correlation function of a random process whose power spectral density is given by UNIT IV 8 (a) Briefly explain the concept of cross power density spectrum. (b) Find the cross correlation of functions sin ωt and cos ωt. 9 (a) The power spectral density of a stationary random process is given by S XX (ω) = A; -k < ω < k = 0; otherwise Find the auto correlation function. (b) Discuss the properties of power spectral density. UNIT V 10 (a) A Gaussian random process X (t) is applied to a stable linear filter. Show that the random process Y(t) developed at the output of the filter is also Gaussian. (b) Discuss about cross correlation between the input X (t) and output Y (t). 11 (a) Derive the relation between PSDs of input and output random process of an LTI system. (b) The input voltage to an RLC series circuit is a stationary random process X(t) with E[X (t)] = and R XX (τ) = 4 + exp (- ). Let Y (t) is the voltage across capacitor. Find E[Y(t)]. Page of

16 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 013/14 PROBABILITY THEY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE) Time: 3 hours Max Marks: 70 1 (a) Define the following with example: (i) Sample space. (ii) Event. (iii) Mutually exclusive event. (iv) Independent event. (v) Exhaustive event. (b) When a die is tossed find the probabilities of the event A = {odd number shown up}; B = {Number larger than 3 shown up} then find out A B and A B. (a) Define probability density function and explain with an example and write its properties. (b) The random variable has following density function: (i) Find value of k. (ii) P ( k k k 3k 3 (a) The density function of a random variable X is: 1 (b) (i) E [x], (ii) E [(x-1) ], (iii) E [3x -1]. For Poisson distributions find out moment generating function and characteristic function. 4 (a) Distinguish between joint distribution and marginal distribution. (b) Joint probability density function of two random variables X and Y. Find: (i) Value of a ; (ii). 5 (a) Explain relation between marginal and joint characteristic function. (b) In a control system a random voltage is known to have mean value and a second moment If the voltage x is amplified by an amplifier that gives an output, and 6 (a) State the conditions for wide sense stationary random process. (b) Explain the classifications of random process. 7 (a) What is an ergodic random process, present the necessary expression to support the argument? (b) Consider a random process where is a random variable uniformly distributed over where is any real number find 8 (a) Explain the relation between power spectrum and auto correlation function of random process. (b) Write any two properties of cross power density spectrum.

17 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 013/14 PROBABILITY THEY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE) Time: 3 hours Max Marks: 70 1 (a) State and prove Baye s theorem of probability. (b) Two similar boxes A and B contain white and 3 red balls 4 white and 5 red balls respectively. If a ball is selected at random from one of the boxes, then find the probability that the box is B when the ball is red. (a) Write the method for defining conditional event. (b) Check whether following is a probability density function or not 3 (a) Explain variance and skew. (b) The mean and variance of binomial distribution are 4 and 4/3 respectively find (c) Find the expected value of the number on a die when thrown. 4 (a) Define and explain the conditional properties. (b) The joint probability density function of two random variables x and y given by (i) Find the value of c. (ii) Marginal distribution function X and Y. 5 (a) Write the expression for expected value of a function of random variables and prove that the mean value of a weighted sum of random variables equals the weighted sum of mean values. (b) Two Gaussian random variables X 1 and X defined by the mean and co-variance matrices two new random variable s Y 1 and Y are formed using the transformation. Find the matrices, and also find the torrelation co-efficient of and i.e.,. 6 (a) Explain the types of statistical averages. (b) Define a random process where A is Gaussian random variable with zero mean and variance is X(t) is stationary in any sense. 7 (a) Define cross correlation and write its properties. (b) A random process is defined as where is a uniform random variable over (0, Verify the process is ergodic in the mean sense and auto correlation sense. 8 (a) Write the relation between cross power spectrum and cross correlation function. (b) If power spectral density of a random process is given by Find the auto correlation function.

18 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 013/14 PROBABILITY THEY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE) Time: 3 hours Max Marks: 70 1 (a) What is sample space? Explain the discrete and continuous sample space. (b) In a box there are 100 resistors having resistances and tolerances as shown in the table. Resistance Tolerance 5% 10% Total Total Let a resistor be selected from the box and assume each resistor are equally likely occur. Define three events A as draw a 47 Ω resistor, B as draw a resistor with 5% tolerance and C as draw a 100 Ω resistor. (i) The probability of drawing a 47 Ω resistor given that the resistor drawn is 5%. (ii) The probability of drawing a 47 Ω resistor given that the resistor drawn is 100 Ω. (iii) The probability of drawing a resistor of 5% tolerance given resistor is 100 Ω. (iv) Find remaining conditional probabilities. (a) Sketch probability density function and probability distribution function of: (i) Exponential distribution. (ii) Rayleigh distribution. (iii) Uniform distribution. (b) Define conditional distribution function and write their properties. 3 (a) State and prove any three properties of variance of a random variable. (b) For binomial density prove: (i). (ii). (iii) = npq. 4 (a) Define and explain joint distribution function and joint density function of two random variables X and Y. (b) The joint space for two random variable X and Y and its corresponding probabilities are shown in table 1, 1, 3, 3 4, Find and plot (a). Marginal distribution of X and Y. (c) (d). Contd. in Page 3 Page 1 of

19 3 5 (a) Show that the variance of a weighted sum of uncorrelated random variables equals the weighted sum of the variance of the random variables. (b) Two random variables X and Y have the density function: (i) Find all the order moment. (ii) Find covariance. (iii) X and Y and uncorrelated. 6 (a) Explain the concept of random process. (b) Distinguish between: (i) Deterministic random process and non-deterministic random process. (ii) Stationary and non- stationary random process. 7 (a) S.T Where A is normal random variable with zero mean and unity variance and uniformly distributed is Assume A and are independent random variable. (b) Discuss Gaussian random process and state its properties. 8 (a) State and explain four properties of power density spectrum of a random process. (b) If find the spectral density function, where a and b are constant. Page of

20 B.Tech II Year I Semester (R09) Regular & Supplementary Examinations December/January 013/14 PROBABILITY THEY AND STOCHASTIC PROCESSES (Common to EIE, E.Con.E & ECE) Time: 3 hours Max Marks: 70 1 (a) Give the classical and axiomatic definitions of probability. (b) A card is drawn from a well shuffled pack of playing cards. What is the probability that in either a spade or ace? (c) What is the probability: (i) Leap year selected at random will contain 53 Sundays? (ii) Non leap year selected at random will contain 53 Sundays? (a) Define random variable and explain types of random variable. (b) In an experiment of rolling a die and flipping a coin the random variable ( is chosen such that. (i) A coin tail (H) outcome corresponds to positive value of that are equal in magnitude to twice the number that shown on die. Map the elements of random variable into points on the real line and explain. 3 (a) Write about transformation. (b) Let x be a random variable defined by density function: 4 Find. 4 (a) Explain method of finding the distribution and density function for a sum of statistically independent random variables. (b) Find constant b: Is a valid joint density function. 5 (a) Two Gaussian random variable and have variance = 4 respectively and correlation coefficient rotation by an angle results in new random variable Y 1 and Y are uncorrelated what is (b) Prove the mean value of weighted sum of random variables equal to the weighted sum of mean sum of mean value. 6 (a) A random process is defined by where A is a continuous random variable uniformly distribution on (0, 1) determine if it is wide sense stationary. (b) Explain the classification of random process with neat sketch. 7 (a) Write and explain the properties of auto correlation wide sense stationary random process. (b) If the random process X(t) had no periodic components and if X(t) is non-zero mean then: Lim 8 (a) For a random process x(t) derive expression for power density spectrum. (b) A random noise X(t) having power spectrum is applied to a network for which h(t) = The network response is denoted by Y(t): (i) Find the average power of X (t). (ii) Find the power spectrum of Y(t). (iii) Find the average power of Y (t).

21 1 B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 01 PROBABILITY THEY & STOCHASTIC PROCESSES (Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering) Time: 3 hours Max. Marks: (a) Explain the following: (i) Random experiment (ii) Trial (iii) Event (iv) sample space. (b) Find the probability of obtaining 14 with 3 dice using Baye s theorem.. (a) Explain with an example discrete, continuous and mixed random variables. (b) Explain CDF with its properties. 3. What is a characteristic function? Explain its properties with its proofs. 4. (a) State and explain Central Limit Theorem. (b) The joint pdf of two random variable X and Y is given by f XY (x, y) = K, (x + y); x = 0,1,, y = 1,,3,4 0 otherwise Find (i) The K value (ii) P(X = 1, Y = ) (iii) P(X 1, Y 3). (iv) f X (x)& f Y (y) (v) f Y X y 1 & f X Y x. 5. (a) Explain about joint moments about the origin with an example. (b) X is a random variable with mean 4 and variance 3. Another random variable Y is related to X as Y=X+7. Determine (i) E[X ] (ii) E[Y] (iii) var [Y] (iv) R XY 6. (a) Differentiate WSS & SSS. (b) Prove the following: (i) R XX (τ) R XX (0).(ii) R XX ( τ) = R XX (τ). (iii) R XX (0) = E[X (t)]. 7. (a) What is meant by co-variance and explain its properties. (b) A random process X(t) = Acosω 0 t + Bsinω 0 t, where ω 0 is constant and A & B are random variables. If A and B are uncorrelated zero mean having same variance σ but different density functions then show that X(t) is a wide sense stationary. 8. (a) Give the relation between cross power spectrum and cross correlation function. (b) A random process has a power spectrum S XX (ω) = 4 ω, 9 ω 6 0 Find (i). Average power (ii) RMS bandwidth. elsewhere

22 B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 01 PROBABILITY THEY & STOCHASTIC PROCESSES (Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering) Time: 3 hours Max. Marks: (a) Explain the following: (i) Principles of counting (ii) probability as a relative frequency. (b) How many positive integers less than 1000 have no common factor with 1000?. (a) Define a random variables and give the conditions for a function to be a random variable. (b) Explain about normal distribution with its properties. 3. Find the mean and variance of Binomial Distribution and Poisson Distribution. 4. (a) Explain about joint density function with its properties. (b) The joint density function of two random variables X ad Y is given by f XY (x, y) = 1 π 3 e 3 (x xy +y ). Determine the marginal probability density function f X (x) and f Y (y). 5. (a) Explain about jointly Gaussian Random variables. (b) A random variable Z has pdf f Z (z) = ae a(z b) u(z b). Show that the characteristic function of z is z (ω) = a a jw e jwb has the probability function P(x) = 1 3 x x = 1,,3 N., 6. (a) What is random process and classify it and explain. (b) A stationary continuous random process X is differentiable and X (t) is its derivates. Show that E X (t) = 0 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) If X(t) is WSS, find R YY (τ) and hence S YY (ω) in terms of S XX (ω) for the product device shown in below fig. x(t) y(t) S XX (ω) product S YY (ω) Acos ωt 8. If a random process X(t) = A 0 (cos ω 0 t + θ), where A 0 & ω 0 are constants and θ is a uniformly distributed random variable in the interval (0, π) Find (i) Whether X(t) is WSS process? (ii) Power in X(t) by time averaging of its second moment. (iii) The power spectral density ofx(t).

23 3 B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 01 PROBABILITY THEY & STOCHASTIC PROCESSES (Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering) Time: 3 hours Max. Marks: (a) Explain the following: (i) probability (ii) Axioms. (b) From the urn containing n balls any numbers of balls are drawn. Show that the probability of drawing an even number of ball is (n 1 1) ( n 1). (a) Differentiate pmf and pdf. (b) Show that f X (x) = 1 σ π ) e(x μ σ, α 0 < xx < α 0, σ > 0 is a distribution function. 3. (a) The pdf of the random variable X follows f X (x) = 1 θ e x θ θ, α 0 < xx < α 0 Find m.g.f. Hence or otherwise find E(X) and var(x). 4. Joint pdf f XY (x, y) of two continuous random variables X and Y is given by f XY (x, y) = K, e (x+y) for x, y 0; 0 otherwise Where K is constant. (i) find K value and f X (x) and f Y (y). (ii) Are X and Y are statistically independent? (iii) Determine joint CDF and marginal distribution function. (iv) Determine the conditional density functions. 5. (a) Explain about joint central moments with an example. (b) A random variable Z with pdf f Z (z) = 1 ; 1 Z 1. Another random variable random variables (RV)s X = Z and Y = Z. Show that X and Y are uncorrelated. 6. (a) What is meant by stochastic process and classify with an example to each. (b) Check the following for WSS. (i) R XX (t, t + τ) = cosst e t+τ ) (ii) R XX (t, t + τ) = sinτ/(1 + τ ) (iii) R XX (t, t + τ) = 10 τ (iv) R X (t, t + τ) = 5e τ 7. (a) Explain in detail the cross power spectral density. (b) If X(t) and Y(t) are random processes. Prove that (i). S XY (ω) = S YX ( ω) = S YX (ω) (ii) S XY (ω) = S YX (ω) if X(t) & Y(t) are uncorrelated WSS random processes. 8. (a) Give the relation between power spectrum and auto correlation function. (b) Find the cross correlation function corresponding to the cross power spectrum S XY = 6/[(9 + ω )(3 + jω) ]

24 4 B.TECH II Year I Semester (R09) Regular & Supplementary Examinations November 01 PROBABILITY THEY & STOCHASTIC PROCESSES (Common to Electronics & Instrumentation Engineering, Electronics & Control Engineering, and Electronics & Communication Engineering) Time: 3 hours Max. Marks: (a) State the Baye s theorem and prove it. (b) There are 300 students in a class room. It is known that 180 can program JAVA, 10 in C++ 30 in SQL, 1 in JAVA and SQL, 18 in C++ and SQL, 1 in JAVA and C++ and 6 in all three languages. (i) A student is selected at random. What is the probability that she can program in exactly two languages. (ii) Two students are selected at random. What is the probability that they can (a) Both program in JAVA (b) Both program only in JAVA.. (a) Explain about pdf. (b) The diameter of a cable X is taken to be a random variable with pdf f X (x) = 6x(1 x), 0 x 1 (i) verify f X (x) is a pdf or not.(ii) Determine b such that P(x < bb) = P(x > bb). 3. (a) Define and explain moments of a random variable. (b) Find the moment generating function of a random variable. 4. (a) Find the density of W=X+Y, where the densities of X and Y to be f X (x) = u(x) u(x 1)& f Y (y) = u(y) u(y 1). (b) Explain the following (i) Joint distribution function. (ii) conditional distribution function (iii) Marginal distribution function. 5. (a) Show that the variance of variance of a weighted sum of uncorrelated random variables equals the weighted sum of t he variances of the random variables. (b) Explain about transformation of multiple random variables. 6. Explain about the concept stationarity in detail connected with stochastic processes. 7. (a) Explain about WSS and prove any two properties of it. (b) Y(t) = X(t) cos(ω 0 t + θ), where X(t) is a random process and θ is uniformly distributed over the interval (0,π). Determine under what conditions is Y(t) wide seuce stationary. Assume θ and X(t) are statistically independent and ω 0 is constant. 8. (a) Write different types of band pass processes with band limited processes. (b) Find the rms band width of the power spectrum Acos πω S XX (ω) = W, ω W 0, ω W Where ω > 0 & WW > 0 are constants?