Engineering Mathematics 08 SUBJECT NAME : Probability & Random Processes SUBJECT CODE : MA645 MATERIAL NAME : University Questions REGULATION : R03 UPDATED ON : November 07 (Upto N/D 07 Q.P) (Scan the above Q.R code for the direct download of this material) Name of the Student: Branch: Unit I (Random Variables) Problems on Discrete & Continuous R.Vs. A random variable X has the following probability distribution. Find: X 0 3 4 5 6 7 P(x) 0 k k k 3k k k 7k k () The value of k () P(.5 X 4.5 / X ) and (3) The smallest value of n for which P( X n). (N/D 00),(M/J 0),(M/J 04). Show that for the probability function, x,,3... p( x) P( X x) x x EX ( ) does not exist. (N/D 0) 0, otherwise 3. The probability function of an infinite discrete distribution is given by P( X j) ( j,,3,...) Find j () Mean of X () PX ( is even) and (3) PX ( is divisible by 3) (N/D 0) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page
Engineering Mathematics 08 4. The probability mass function of random variable X is defined as P( X 0) 3 C, P( X ) 4C 0C r 0,,. Find () The value of C, P( X ) 5C () P(0 X / x 0) (3) The distribution function of X, where C 0 and P( X r) 0 if (4) The largest value of X for which F( x). (A/M 00) 5. A random variable X has pdf x kx e, x 0 f( x). Find the rth moment of 0, otherwise X about origin. Hence find the mean and variance. (M/J 03) 6. The probability density function of a random variable X is given by x, 0 x f X ( x) k( x), x. 0, otherwise () Find the value of k. () Find P(0. x.) (3) What is P0.5 x.5 / x (4) Find the distribution function of f( x ). (A/M 0) x, 0 x b 7. The pdf of a random variable X is given by f( x). For what value 0, otherwise of b is f( x ) a valid pdf? Also find the cdf of the random variable X with the above pdf. (N/D 05) k 8. A continuous R.V. X has the p.d.f. ( ), x f x x. Find 0, elsewhere () the value of k () Distribution function of X (3) PX ( 0) (N/D 0) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page
Engineering Mathematics 08 Moments and Moment Generating Function. Find the MGF of the two parameter exponential distribution whose density function is given by ( x a) f ( x) e, x a and hence find the mean and variance. (A/M 00). Describe Binomial B( n, p ) distribution and obtain the moment generating function. Hence compute () the first four moments and () the recursion relation for the central moments. (M/J 04) 3. Find the moment generating function of Poisson distribution and hence find its mean and variance. (A/M 0),(N/D 05) 4. Describe gamma distribution. Obtain its moment generating function. Hence compute its mean and variance. (M/J 03),(N/D 07) 5. Define the moment generating function (MGF) of a random variable. Derive the MGF, mean, variance and the first four moment of a Gamma distribution. (M/J 04) 6. Find the mean and variance of Gamma distribution. (N/D 03),(N/D 04) 7. Find the nth moment about mean of normal distribution. (N/D 04) 8. If the probability density of X is given by moment. Hence evaluate E X ( x) for 0 x f( x), find its rth 0, otherwise. (N/D 0) 9. Find the M.G.F. of the random variable X having the probability density function x x e, x 0 f( x) 4. Also deduce the first four moments about the origin. 0, elsewhere (N/D 00),(M/J 0) x e, x 0 0. A random variable X has the pdf f( x). Obtain the mgf and first four 0, x 0 moment about the origin. Find mean and variance of the same. (N/D 04) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 3
Engineering Mathematics 08 x, x 0 x. Find the MGF of the random variable X having the pdf f( x) 4e. 0, elsewhere Also deduce the first four moments about the origin. (M/J 04). Find MGF corresponding to the distribution e, 0 f ( ) and hence find 0, otherwise its mean and variance. (N/D 0) Problems on Distributions. Derive Poisson distribution from binomial distribution. (N/D 03),(N/D 04). The number of monthly breakdowns of a computer is a random variable having a Poisson distribution with mean equal to.8. Find the probability that this computer will function for a month () without a breakdown () with only one breakdown. (N/D 07) 3. If the probability that a target is destroyed on any one shot is 0.5. What is the probability that it would be destroyed on 6 th attempt? (N/D 07) 4. If the probability that an applicant for a driver s license will pass the road test on any given trial is 0.8. What is the probability that he will finally pass the test () On the fourth trial and () In less than 4 trials? (A/M 00) 5. State and prove memory less property of Geometric distribution. (N/D 05) 6. A random variable X is uniformly distributed over (0,0). Find 3, 7, 5 and 7 P X P X P X P X. (M/J 03) 7. The time in hours required to repair a machine is exponentially distributed with parameter /. () What is the probability that the repair time exceeds hours? () What is the conditional probability that a repair takes atleast 0 hours given that its duration exceeds 9 hours? (M/J 0) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 4
Engineering Mathematics 08 8. The marks obtained by a number of students in a certain subject are assumed to be normally distributed with mean 65 and standard deviation 5. If 3 students are selected at random from this group, what is the probability that two of them will have marks over 70? (A/M 00),(A/M 0) 9. In a test on 000 electric bulbs, it was found that the life of a Philips bulbs was normally distributed with an average of 400 hours and S.D. of 60 hours. Estimate the number of bulbs likely to burn for (i) more than 50 hours, (ii) less than 950 hours. (N/D 07) 0. Assume that the reduction of a person s oxygen consumption during a period of Transcendental Meditation (T.M) is a continuous random variable X normally distributed with mean 37.6 cc/mm and S.D 4.6 cc/min. Determine the probability that during a period of T.M. a person s oxygen consumption will be reduced by () at least 44.5 cc/min () at most 35.0 cc/min (3) any where from 30.0 to 40.0 cc/mm. (N/D 0). In a normal distribution, 3% of items are under 45 and 8% of items are over 64. Find the mean and the standard deviation of the distribution. (N/D 05). Let X and Y be independent normal variates with mean 45 and 44 and standard deviation and.5 respectively. What is the probability that randomly chosen values of X and Y differ by.5 or more? (N/D 0) 3. Given that X is distributed normally, if PX ( 45) 0.3 and PX ( 64) 0.08, find the mean and standard deviation of the distribution. (M/J 0) 4. If X and Y are independent random variables following (8, ) N and N,4 3 respectively, find the value of such that PX Y P X Y Unit II (Two Dimensional Random Variables) Joint distributions Marginal & Conditional. (N/D 00). The joint probability mass function of XY, is given by p x, y k x 3y, x 0,, ; y,,3. Find k and all the marginal and conditional probability distributions. Also find the probability distribution of X Y. (N/D 03),(N/D 04) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 5
Engineering Mathematics 08. The bivariate probability distribution of XY, given below: Y 3 4 5 6 X 0 0 0 /3 /3 /3 3/3 /6 /6 /8 /8 /8 /8 /3 /3 /64 /64 0 /64 Find the marginal distributions, conditional distribution of X given Y = and conditional distribution of Y given X = 0. (A/M 00), 3 ; 7 x. Find all the marginal and conditional probability function of X 3. The joint probability mass function of XY, is given by p x y x y x 0,, and,,3 and Y. (N/D 05) 8 f ( x, y) xy, 9 0 x y and f ( x, y) 0, otherwise. Find the densities of X and Y, and the 4. The joint p.d.f of two dimensional random variable XY, is given by conditional densities f ( x / y) and f ( y / x ). (A/M 00) 5. The two dimensional random variable XY, has the joint density function x 7y f ( x, y), x 0,,, y 0,,. Obtain f ( y / x ) and f ( x / y ). (N/D 07) 7 6. The joint probability density function of random variable X and Y is given by 8xy, x y f ( x, y) 9. Find the conditional density functions of X and Y. 0, otherwise 7. The joint pdf of a two-dimensional random variable XY, is given by (N/D 0) x f ( x, y) xy, 0 x,0 y. Compute PY ( / ), 8 P( X / Y / ) and P( X Y ). (N/D 0) 8. Given the joint pdf of X and Y Cx( x y), 0 x, x y x f ( x, y). 0, otherwise () Evaluate C () Find the marginal pdf of X (3) Find the conditional density of / Y X. (M/J 03) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 6
Engineering Mathematics 08 9. The joint pdf of XY, is ( x y) f ( x, y) e ; x, y 0. Are X and Y independent? (N/D 05) 0. If the joint pdf of two dimensional random variable XY, is given by xy x, 0 x ; 0 y f ( x, y) 3. Find (M/J 04) 0, otherwise (i) P X (ii) P Y X (iii) P X Y (iv) Find the conditional density functions. Covariance, Correlation and Regression. The joint pdf of the random variable XY, is f ( x, y) 3 x y 0 x, 0 y, x y find,, Cov X Y. (M/J 04). Find the covariance of X and Y, if the random variable (X,Y) has the joint p.d.f f ( x, y) x y, 0 x, 0 y and f ( x, y) 0, otherwise. (A/M 00),(N/D 07) 3. The joint probability density function of random variable XY, is given by x y f ( x, y) Kxye, x 0, y 0. Find the value of K and, Cov X Y. Are X and Y independent? (M/J 0) 4. The joint pdf of a random variable XY, is 5 y f ( x) 5 e ; 0 x 0., y 0. Find the covariance of X and Y. (N/D 05) 5. The joint probability density function of the two dimensional random variable XY, is x y, 0 x, 0 y f ( x, y). Find the correlation coefficient 0, otherwise between X and Y. (N/D 0) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 7
Engineering Mathematics 08 6. Two random variables X and Y have the joint probability density function given by f XY k( x y), 0 x, 0 y ( x, y). (N/D 00) 0, otherwise () Find the value of k () Obtain the marginal probability density functions of X and Y. (3) Also find the correlation coefficient between X and Y. 7. Two independent random variables X and Y are defined by 4 ax, 0 x 4 by, 0 y fx ( x) and fy (y). Show that 0, otherwise 0, otherwise U X Y and V X Y are uncorrelated. (M/J 03) 8. If X and Y are uncorrelated random variables with variances 6 and 9. Find the correlation co-efficient between X Y and X Y. (M/J 0) 9. If the independent random variables X and Y have the variances 36 and 6 respectively, find the correlation coefficient between( X Y) and ( X Y). (N/D 0) 0. The equations of two regression lines are 3x y9 and 3y9x 46. Find x, y and Correlation Coefficient between X and Y. (M/J 03). The regression equation of X on Y is 3Y 5X08 0. If the mean value of Y is 44 and the variance of X is 9/6 th of the variance of Y. Find the mean value of X and the correlation coefficient. (A/M 0). Marks obtained by 0 students in Mathematics ( x ) and Statistics ( y ) are given below: x : 60 34 40 50 45 40 43 4 64 y : 75 3 33 40 45 33 30 34 5 Find the two regression lines. Also find y when x 55. (M/J 04) Transformation of the random variables. If X and Y are independent random variables with density function f Z X, x ( x) and 0, otherwise y, y 4 fy ( y) 6, find the density function of 0, otherwise XY. (A/M 0) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 8
Engineering Mathematics 08. X and Y are independent with a common PDF (exponential): x e, x 0 f( x) and 0, x 0 y e, y 0 f( y). Find the PDF for X Y. (N/D 0) 0, y 0 3. The random variables X and Y each follow exponential distribution with parameter and are independent. Find the pdf of U X Y. (N/D 05) x y 4. If X and Y are independent RVs with pdf s e, x 0and e, y 0respectively, find X the pdfs of U X and V X Y. Are U and V independent? (N/D 03) Y 5. The waiting times X and Y of two customers entering a bank at different times are x e, x 0 assumed to be independent random variables with p.d.fs f( x) and 0, otherwise y e, y 0 f( y). Find the joint p.d.f of U X Y, V 0, otherwise X. (N/D 07) X Y 6. If X and Y are independent random variables with probability density functions f x e x 4 x X ( ) 4, 0; f y e y respectively. (N/D 0) y Y ( ), 0 X (i) Find the density function of U, V X Y X Y (ii) Are U and V independent? (iii) What is PU 0.5? 7. Let XY, be a two dimensional random variable and the probability density function be given by f ( x, y) x y, 0 x, y. Find the p.d.f of U XY. (M/J 0) 8. If X and Y are independent RVs each normally distributed with mean zero and variance, find the pdf of R X Y and Y tan X. (N/D 03) 9. If X and Y are independent continuous random variables, show that the pdf of U X Y is given by h( u) f x( v) f y( u v) dv. (N/D 00) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 9
Engineering Mathematics 08 Unit III (Random Processes) Verification of SSS and WSS process. Examine whether the random process X( t) Acos( t ) is a wide sense stationary if A and are constants and is uniformly distributed random variable in (0,π). (A/M 00),(N/D 0),(N/D 07). A random process Xt () defined by X( t) Acos t Bsin t, t, where A and B are independent random variables each of which takes a value with probability / 3 and a value with probability / 3. Show that Xt () is wide sense stationary. (A/M 0),(M/J 03),(N/D 05) 3. Prove that the random processes Xt () and Yt () defined by X( t) Acost Bsint and Y( t) Bcost Asint are jointly wide sense stationary. (M/J 04) 4. If the two RVs Ar and Br are uncorrelated with zero mean and, show that the process x( t) A cos t B sin t E A E B r r r n is r r r r r wide-sense stationary. (N/D 03),(N/D 04) 5. The process Xt () whose probability distribution under certain condition is given by n ( at), n,... n ( at) PX() t n. Find the mean and variance of the process. at, n 0 at Is the process first-order stationary? (N/D 00),(N/D 0),(N/D 0),(M/J 04),(N/D 04) 6. If () Xt is a WSS process with autocorrelation R( ) Ae, determine the second order moment of the RVX(8) X(5). (M/J 0) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 0
Engineering Mathematics 08 Problems on Markov Chain. The transition probability matrix of a Markov chain Xt (), n,,3,... having three 0. 0.5 0.4 states,,3 is P 0.6 0. 0., and the initial distribution is 0.3 0.4 0.3 (0) P 0.7 0. 0., Find P X and P X, X 3, X 3, X 3. 3 0 (A/M 00). A man either drives a car or catches a train to go to office each day. He never goes two days in a row by train. But he drives one day, then the next day is just as likely to drive again as he is to travel by train. Now suppose that on the first day of the week, the man tossed a fair dice and drove to work if and only if a 6 appeared. Find the probability that he takes a train on the fourth day and the probability that he drives to work on the fifth day. (N/D 05),(N/D 07) Poisson Process. If the process X( t); t 0 is a Poisson process with parameter, obtain P X() t n. Is the process first order stationary? (N/D 00),(N/D 0),(M/J 04). State the postulates of a Poisson process and derive the probability distribution. Also prove that the sum of two independent Poisson processes is a Poisson process. (N/D 0) 3. Define a Poisson process. Show that the sum of two Poisson processes is a Poisson process. (M/J 03),(N/D 03) 4. A Hard Disk fails in a computer system and it follows a Poisson distribution with mean rate of per week. Find the probability that weeks have elapsed since last failure. If we have 5 extra hard disks and the next supply is not due i n 0 weeks, find the probability that the machine will not be out or order in next 0 weeks.(n/d 07) 5. If customers arrive at a counter in accordance with a Poisson process with a mean rate of per minute, find the probability that the interval between consecutive arrivals is () more that minute () between minute and minute and (3) 4 min. or less. (M/J 0) 6. Assume that the number of messages input to a communication channel in an interval of duration t seconds, is a Poisson process with mean 0.3. Compute () The probability that exactly 3 messages will arrive during 0 second interval Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page
Engineering Mathematics 08 () The probability that the number of message arrivals in an interval of duration 5 seconds is between 3 and 7. (A/M 00) 7. Suppose that customers arrive at a bank according to a Poisson process with a mean rate of 3 per minute. Find the probability that during a time interval of min. () exactly 4 customers arrive and () more than 4 customers arrive (N/D 03) (3) Fewer than 4 customers arrive (N/D 05) 8. Prove that the interval between two successive occurrences of a Poisson process with parameter has an exponential distribution with mean. (A/M 0) Normal (Gaussian) & Random telegraph Process. If () Xt is a Gaussian process with ( t) 0 probability that C t, t 6e t t and, find the () X(0) 8 () X(0) X(6) 4 (A/M 0),(N/D 03),(N/D 04). Suppose that Xt () is a Gaussian process with, x Rxx 5e 0.. Find the probability that X(4). (M/J 0) 3. Define a semi random telegraph signal process and prove that it is evolutionary. (M/J 03),(N/D 05) 4. Define random telegraph signal process and prove that it is wide-sense stationary. (N/D 03),(N/D 07) 5. Define a semi random telegraph signal process and random telegraph signal process and prove also that the former is evolutionary and the latter is wide-sense stationary. (N/D 04) 6. Prove that a random telegraph signal process Y( t) X( t) is a Wide Sense Stationary Process when is a random variable which is independent of Xt (), assume value t t and with equal probability and R ( t, t ) e. (N/D 00),(N/D 0),(M/J 04) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page
Engineering Mathematics 08 Unit IV (Correlation and Spectral densities) Auto Correlation from the given process. Find the autocorrelation function of the periodic time function of the period time function X( t) Asint. (A/M 00). Find the mean and auto correlation of the Poisson process. (M/J 04) Relationship between R and S. Define spectral density of a stationary random process Xt (). Prove that for a real random process Xt (), the power spectral density is an even function. (M/J 03). The autocorrelation function of the random binary transmission Xt () is given by R( ) for T and R( ) 0 for T. Find the power spectrum of the T process Xt (). (A/M 00) 3. Find the power spectral density of the random process whose auto correlation function, for is R( ). (N/D 00),(N/D 0),(N/D 07) 0, elsewhere 4. Find the power spectral density function whose autocorrelation function is given by A R cos 0. (M/J 0) 5. A random process () Xt is given by X( t) Acos pt Bsin pt, where A and B are independent random variables such that E( A) E( B) 0 and E( A ) E( B ). Find the power spectral density of the process. (N/D 04) 6. The autocorrelation function of a random process is given by R( ) ; ;. Find the power spectral density of the process. (N/D 0) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 3
Engineering Mathematics 08 7. The autocorrelation function of a random process is given by R( ) S ; ; ( ) 4 sin. Prove that its spectral density is. (N/D 03) 8. The Auto correlation function of a WSS process (random telegraph single process) is given by R( ) e, determine the power spectral density of the process (random telegraph single process). (A/M 0),(N/D 05) 9. Find the power spectral density of a WSS process with autocorrelation function R( ) e. (N/D 04) 0. The autocorrelation function of the random telegraph signal process is given by e R( ). Determine the power density spectrum of the random telegraph signal. (N/D 03). Find the power spectral density of a WSS process Xt () which has an autocorrelation R ( ) /, xx A0 T T t T. (N/D 0). Find the autocorrelation function of the process Xt () for which the power spectral density is given by S ( ) for and S ( ) 0 for.(a/m 00) 3. The power spectral density function of a zero mean WSS process Xt () is given by, 0 S( ). Find R( ) and show that Xt () and Xt are 0, otherwise 0 uncorrelated. (A/M 0) 4. If the power spectral density of a WSS process is given by b a S( ) a 0,, a. Find the autocorrelation function of the process. a (N/D 03),(N/D 04),(N/D 07) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 4
Engineering Mathematics 08 5. The power spectrum of a WSS process Xt () is given by S( ). Find its auto correlation function R( ). (N/D 05) Relationship between R and S XY XY. The cross-correlation function of two processes Xt () and Yt () is given by AB RXY ( t, t ) sin( 0 ) cos 0 t where ABand, 0 are constants. Find the cross-power spectrum SXY ( ). (M/J 0). The cross power spectrum of real random processes Xt () and Yt () S xy is given by a bj, for ( ). Find the cross correlation function. 0, elsewhere (N/D 00),(A/M 0),(N/D 0) 3. If the cross power spectral density of Xt () and Yt () is S XY ib a,, 0 ( ) where a and b are constants. Find the 0, otherwise cross correlation function. (M/J 03),(N/D 07) 4. Two random processes Xt () and Y( t) are defined as follows: Xt ( ) Acos( t ) and Y( t) Bsin( t ) where A, Band are constants; is a uniform random variable over 0,. Find the cross correlation function of Xt () and Y( t ). Properties, Theorems and Special problems (M/J 03),(N/D 05). State and prove Weiner Khintchine Theorem. (N/D 00),(A/M 0),(N/D 0),(N/D0),(M/J 03),(M/J 04). If () Xt and () Yt are two random processes with auto correlation function R ( ) and RYY ( ) respectively then prove that RXY ( ) R (0) RYY (0). Establish any two properties of auto correlation function R ( ).(N/D 00),(N/D0) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 5
Engineering Mathematics 08 3. Given the power spectral density of a continuous process as S 9 4 5 4. Find the mean square value of the process. (N/D 0) 4. A stationary random process Xt () with mean has the auto correlation function R ( ) 4 0 e. Find the mean and variance of Y X( t) dt. (M/J 0) 5. The random binary transmission process Xt () is a WSS process with zero mean and autocorrelation function R( ), where T is a constant. Find the mean and T variance of the time average of Xt () over 0,T. Is Xt () mean ergodic? 6. Xt () and Yt () 0 (N/D 04) are zero mean and stochastically independent random processes having autocorrelation functions R ( ) e and R ( ) cos respectively. Find () The autocorrelation function of W( t) X( t) Y( t) and Z( t) X( t) Y( t) () The cross correlation function of W() t and Zt (). (A/M 00) YY 7. Let Xt () and Yt () be both zero-mean and WSS random processes Consider the random process Zt () defined by Z( t) X( t) Y( t). Find () The Auto correlation function and the power spectrum of Zt () if Xt () and Yt () are jointly WSS. (N/D 07) () The power spectrum of Zt () if Xt () and Yt () are orthogonal. (M/J 0) 8. If Y( t) X( t a) X( t a), prove that R ( ) R ( a) R ( a). Hence prove that 9. If the process () YY ( ) 4sin ( ) YY S a S. (N/D 05) Xt is defined as X( t) Y( t) Z( t) independent WSS processes, prove that () R R R and xx yy zz () where Yt () and Z( ) Sxx S yy Szz d (N/D 03) t are Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 6
Engineering Mathematics 08 Unit V (Linear Systems with Random inputs) Input and Output Process. If the input to a time invariant stable, linear system is a WSS process, prove that the output will also be a WSS process. (N/D 0),(M/J 03). Show that if the input Xt () is a WSS process for a linear system then output Yt () is a WSS process. Also find RXY ( ). (N/D 00),(N/D 0),(M/J 04) 3. For a input output linear system X( t), h( t), Y( t ), derive the cross correlation function RXY ( ) and the output autocorrelation function RYY ( ). (N/D 0) 4. Check whether the following systems are linear () y( t) t x( t) () y( t) x ( t). (N/D 04) 5. Prove that the spectral density of any WSS process is non-negative. (N/D 03) 6. Consider a system with transfer function. An input signal with autocorrelation j function m ( ) m is fed as input to the system. Find the mean and mean-square value of the output. (A/M 0),(M/J 0) 7. If () Xt is a WSS process and if Y( t) h( ) X( t ) d then prove that () R ( ) R ( )* h( ) where * stands for convolution. () 8. If () XY * XY ( ) ( ) ( ) S S H. (M/J 0) Xt is a WSS process and if Y( t) h( u) X( t u) du, prove that: (i) R ( ) R ( ) h( ) XY (ii) R ( ) R ( ) h( ) where denotes convolution YY XY (iii) S ( ) S ( ) H ( ) where H ( ) is the complex conjugate of H( ) XY (iv) S ( ) S ( ) H( ) (N/D 05),(N/D 07) YY Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 7
Engineering Mathematics 08 9. Assume a random process Xt () is given as input to a system with transfer function H( ) for 0. If the autocorrelation function of the input process is 0 N0 () t, find the autocorrelation function of the output process. (A/M 00) 0. If Xt () is the input voltage to a circuit and Yt () is the output voltage. Xt () is a stationary random process with X 0 and R ( ) e. Find the mean Y and power spectrum S ( ) of the output if the system transfer function is given by H( ) YY. (N/D 00),(N/D 0) i. Xt () is the input voltage to a circuit (system) and Yt () is the output voltage. is a stationary random process with x 0 and R xx ( ) e R, if the power transfer function is H yy () Xt. Find y, S yy R. R il (N/D 03),(M/J 04),(N/D 07) Input and Output Process with Impulse Response and t. A system has an impulse response h( t) e U( t), find the power spectral density of the output Yt () corresponding to the input Xt (). (N/D 00),(N/D 0),(M/J 04),(N/D 07). A random process Xt () is the input to a linear system whose impulse function is t h( t) e ; t 0. The auto correlation function of the process is ( ) R e. Find the power spectral density of the output process Yt (). (M/J 03) 3. A random process Xt () is the input to a linear system whose impulse response is t h( t) e, t 0. If the autocorrelation function of the process is R ( ) e, determine the cross correlation function RXY ( ) between the input process Xt () and the output process () Yt and the cross correlation function R ( ) YX between the output process Yt () and the input process Xt (). (N/D 05) 4. A stationary random process Xt () having the autocorrelation function R ( ) ( ) A is applied to a linear system at time t 0 where f ( ) represent the bt impulse function. The linear system has the impulse response of h( t) e u( t) where Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 8
Engineering Mathematics 08 ut () represents the unit step function. Find R ( ). Also find the mean and variance of Yt (). (A/M 0),(M/J 0) 5. A wide sense stationary random process () YY a Xt with autocorrelation R ( ) e where A and a are real positive constants, is applied to the input of an Linear bt transmission input system with impulse response h( t) e u( t) where b is a real positive constant. Find the autocorrelation of the output Yt () of the system.(a/m 00) t 6. A linear system is described by the impulse response RC h( t) e u( t). Assume an RC input process whose Auto correlation function is B( ). Find the mean and Auto correlation function of the output process. (A/M 0) RC 7. A linear system is described by the impulse response h() t e. Assume an RC input signal whose autocorrelation function is B( ). Find the autocorrelation mean and power of the output. (N/D 04) 8. Let Xt () be a WSS process which is the input to a linear time invariant system with unit impulse ht () and output Yt (), then prove that t S ( ) H( ) S ( ) where H( ) is Fourier transform of ht (). (N/D 0),(M/J 03) Band Limited White Noise. A wide sense stationary noise process Nt () has an auto correlation function R ( ) NN 3 Pe where P is a constant. Find its power spectrum. (M/J 03). If Y( t) Acos( 0t ) N( t), where A is a constant, is a random variable with a uniform distribution in, and Nt () is a band-limited Gaussian white noise N0, for 0 B with power spectral density SNN ( ). Find the power 0, elsewhere spectral density Yt (). Assume that Nt () and are independent. yy (N/D 00),(N/D 0),(N/D 03),(M/J 04) 3. If Y( t) Acos( t ) N( t), where A is a constant, is a random variable with a uniform distribution in (, ) and Nt () is a band limited Gaussian white noise with xx Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 9
Engineering Mathematics 08 N0 a power spectral density SNN ( ) for 0 B and SNN ( ) 0, elsewhere. Find the power spectral density of Yt (), assuming that Nt () and are independent. (A/M 00) 4. If Nt () is a band limited white noise centered at a carrier frequency 0 such that S NN 5. If () N ( ), for 0, elsewhere 0 0 B. Find the autocorrelation of () Nt. (A/M 0),(M/J 0) Xt is a band limited process such that S ( ) 0 when, prove that R (0) R ( ) R (0). (A/M 00) 6. A white Gaussian noise Xt () with zero mean and spectral density low-pass RC filter shown in the figure. N 0 is applied to a Determine the autocorrelation of the output Yt (). (N/D 0) ----All the Best---- Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 0