Name of the Student: Problems on Discrete & Continuous R.Vs

Engineering Mathematics 05 SUBJECT NAME : Probability & Random Process SUBJECT CODE : MA6 MATERIAL NAME : University Questions MATERIAL CODE : JM08AM004 REGULATION : R008 UPDATED ON : Nov-Dec 04 (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). The probability mass function of random variable X is defined as P( X 0) 3 C, P( X ) 4C 0C r 0,,, P( X ) 5C. Find () The value of C () 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) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page

Engineering Mathematics 05 3. 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) 4. 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) 5. A continuous R.V. X has the p.d.f. () the value of k k, x f( x) x. Find 0, elsewhere () Distribution function of X (3) PX ( 0) (N/D 0) 6. Show that for the probability function, x,,3... p( x) P( X x) x x EX ( ) does not exist. (N/D 0) 0, otherwise 7. 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 05 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. Derive the m.g.f of Poisson distribution and hence or otherwise deduce its mean and variance. (A/M 0) 4. Describe gamma distribution. Obtain its moment generating function. Hence compute its mean and variance. (M/J 03) 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 ( x) for 0 x f( x), find its rth 0, otherwise E X. (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 05 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). 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) 3. 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) 4. 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) 5. 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 set, what is the probability that exactly of them will have marks over 70? (A/M 0) 6. 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 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 4

Engineering Mathematics 05 (3) anywhere from 30.0 to 40.0 cc/mm. (N/D 0) 7. 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) 8. 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) 9. 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. (N/D 00) 0. 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 at least 0 hours given that its duration exceeds 9 hours? (M/J 0) Function of random variables. If X is uniformly distributed in,, then find the probability density function of X Y sin. (N/D 00). If X is a uniform random variable in the interval @, find the probability density function Y X EY. (N/D 0) and x e, 0 x 3. The random variable X has exponential distribution with f( x). 0, otherwise Find the density function of the variable given by () Y 3X 5 () Y X (N/D 0) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 5

Engineering Mathematics 05 Unit II (Two Dimensional Random Variables) Joint distributions Marginal & Conditional. 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) 8 f ( x, y) xy, 9 0 x y and f ( x, y) 0, otherwise. Find the densities of X and Y, and the. 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) 3. 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 4. 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) 5. 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) 6. 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 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 6

Engineering Mathematics 05 (iii) P X Y (iv) Find the conditional density functions. 7. Find 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) Covariance, Correlation and Regression. The joint pdf of the random variable XY, is f ( x, y) 3 x y 0 y, x y find,, 0 x, 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) 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 probability density function of the two dimensional random variable XY, x y, 0 x, 0 y is f ( x, y). Find the correlation coefficient 0, otherwise between X and Y. (N/D 0) 5. 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. Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 7

Engineering Mathematics 05 (3) Also find the correlation coefficient between X and Y. 6. 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 U X Y 0, otherwise 0, otherwise and V X Y are uncorrelated. (M/J 03) 7. 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) 8. 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) 9. 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) 0. 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. 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). 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 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 8

Engineering Mathematics 05 x y 3. 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 4. 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? 5. 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) 6. If X and Y are independent RVs each normally distributed with mean zero and Y variance, find the pdf of R X Y and tan X. (N/D 03) 7. 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) Central Limit Theorem. State and prove the central limit theorem for iid random variables. (M/J 03),(N/D 03),(N/D 04). A sample of size 00 is taken from a population whose mean is 60 and variance is 400. Using Central Limit Theorem, find the probability with which the mean of the sample will not differ from 60 by more than 4. (A/M 00) 3. The life time of a particular variety of electric bulb may be considered as a random variable with mean 00 hours and standard deviation 50 hours. Using central limit theorem, find the probability that the average life time of 60 bulbs exceeds 50 hours. (A/M 0),(N/D 04) 4. Let X, X, X3,... Xn be Poisson variates with parameter and Sn X X X3... Xn where n 75. Find p0 S n 60 using central limit theorem. (M/J 0) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 9

Engineering Mathematics 05 5. If X, X, X3,... Xn are uniform variates with mean.5 and variance 3 / 4, use CLT to estimate 08.6 p S n where S X X X3... X, n 48. n n (N/D 0) 6. If Vi, i,,3...0 are independent noise voltages received in an adder and V is the sum of the voltages received, find the probability that the total incoming voltage V exceeds 05, using the central limit theorem. Assume that each of the random variables V i is uniformly distributed over (0,0). (N/D 00) Unit III (Classification of 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). 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) 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 E A E B r r r Br are uncorrelated with zero mean and n r r r r is r, show that the process x( t) A cos t B sin t 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 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 0

Engineering Mathematics 05 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) Ergodic processes, Mean ergodic and Correlation ergodic. The random binary transmission process Xt () is a wide sense process with zero mean and autocorrelation function R( ), where T is a constant. Find the mean and T variance of the time average ofxt () over (0, T). Is Xt () mean ergodic?. A random process has sample functions of the form X( t) Acos t (A/M 00), where is constant, A is a random variable with mean zero and variance one and is a random variable that is uniformly distributed between 0 and. Assume that the random variables A and are independent. Is Xt () is a mean ergodic process? (A/M 0) 3. If the WSS process () distributed over,, prove that () Problems on Markov Chain Xt is given by X( t) 0cos(00 t ), where is uniformly Xt is correlation ergodic. (N/D 00),(M/J 0),(N/D 0),(M/J 04). The transition probability matrix of a Markov chainxt (), n,,3,... having three states,,3 is 0. 0.5 0.4 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 Poisson Process 3. 3 0. If the process X( t); t 0 is a Poisson process with parameter, obtain P X() t n (A/M 00). Is the process first order stationary? (N/D 00),(N/D 0),(M/J 04) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page

Engineering Mathematics 05. 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. 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) 5. 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 () The probability that the number of message arrivals in an interval of duration 5 seconds is between 3 and 7. (A/M 00) 6. 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) 7. 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, Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 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)

Engineering Mathematics 05 4. Define random telegraph signal process and prove that it is wide-sense stationary. (N/D 03) 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) 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 is, for R( ). (N/D 00),(N/D 0) 0, elsewhere 4. Find the power spectral density function whose autocorrelation function is given by A R cos 0. (M/J 0) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 3

Engineering Mathematics 05 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. 7. The autocorrelation function of a random process is given by R( ) S ; ; ( ) 4 sin. Prove that its spectral density is 8. The Auto correlation function of a WSS process is given by (N/D 0). (N/D 03) R( ) e determine the power spectral density of the process. (A/M 0) 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 auto correlation 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) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 4

Engineering Mathematics 05 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) 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 () S XY ib a,, 0 ( ) where a and b are constants. Find the 0, otherwise cross correlation function. (M/J 03) 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 ).(M/J 03) Properties, Theorem and Special problems is. 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 05 9 3. Given the power spectral density of a continuous process as S 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 8. If the process () () The Auto correlation function and the power spectrum of Zt () if Xt () and Yt () are jointly WSS. () The power spectrum of Zt () if Xt () and Yt () are orthogonal. Xt is defined as X( t) Y( t) Z( t) independent WSS processes, prove that () R R R and xx yy zz () (M/J 0) where Yt () and Z( ) t are Sxx S yy Szz d (N/D 03) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 6

Engineering Mathematics 05 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. () XY * XY ( ) ( ) ( ) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 7 S S H. (M/J 0) 8. Assume a random process Xt () is given as input to a system with transfer function H( ) for 0 0. If the autocorrelation function of the input process is N0 () t, find the autocorrelation function of the output process. (A/M 00) 9. 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 power spectrum S ( ) of the output if the system transfer function is given by H( ) YY and. (N/D 00),(N/D 0) i

Engineering Mathematics 05 0. 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 and R. (N/D 03),(M/J 04) R il Input and Output process with impulse response 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). 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 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 ut () represents the unit step function. Find R ( ). Also find the mean and variance of Yt (). (A/M 0),(M/J 0) 4. 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 RC 5. A linear system is described by the impulse response 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 6. 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) t Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 8

Engineering Mathematics 05 7. 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 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 with power spectral density S NN yy N0, for 0 B ( ). Find the power 0, elsewhere spectral density Yt (). Assume that Nt () and are independent. (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 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. xx (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) Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9846897) Page 9

Engineering Mathematics 05 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