P 1.5 X 4.5 / X 2 and (iii) The smallest value of n for

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1 DHANALAKSHMI COLLEGE OF ENEINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING MA645 PROBABILITY AND RANDOM PROCESS UNIT I : RANDOM VARIABLES PART B (6 MARKS). A random variable X has the following probability distribution X P(X) 0 K k k 3k k k 7k k P.5 X 4.5 / X and (iii) The smallest value of n for Find (i) The value of k (ii) which P X n [A/M -],[N/D -0,M/J-4] (8). The probability function of an infinite discrete distribution is given by P X j, j,,3,.... Find the mean and variance of the distribution. Also find j P( X is even), P( X 5) and P( X is divisible by 3). (8) x Kx e ; x 0 3. A random variable X has p.d.f f( x). Find the rth moment of X about origin. 0 ; otherwise Hence find the mean and variance. [M/J-3] (8) x 4. The cumulative distribution function of a random variable X is F( x) ( x) e, x 0. Find (i) the probability density function of X (ii) Mean and Variance of X. (8) 5. Derive Poisson distribution is limiting case of binomial distribution. (8) 6. If the probability that an applicant for a drivers licence will pass the road test on any given trial is 0.8, what is the probability that he will finally pass the test (a) on the fourth trial and (b) in fewer than 4 trials? [A/M-0,5] (8)

2 7. The time in hours required to repair a machine is exponentially distributes with parameter. (i) What is the probability that the repair time exceeds h? (ii) What is the (8) conditional probability that a repair takes atleast 0h given that its duration exceeds 9h? [A/M -] 8. The marks obtained by a number of students in a certain subject are assumed to be approximately normally distribution with mean 55 and a S.D of 5. If 5 students are taken at random from this set, what is the probability that three of them would have scored marks above 60? [A/M -0] (8) 9. 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? (8) 0. If X and Y are independent random variables following N(8,) and N(,4 3) respectively, find the value of such that P[X Y ] P[ X Y ]. (8)

3 UNIT II : TO DIMENTIONAL RANDOM VARIABLES PART B (6 MARKS). The two dimensional random variables ( XY, ) has the joint density function x y f ( x, y), x 0,, ; y 0,,. Find (i) the marginal distributions (ii) the 7 conditional distribution of Y given X x (iii) the conditional distribution of X given Y y (iv) the conditional distribution of X given Y. (8). The joint p.d.f of a two dimensional random variable of ( XY, ) is given by x f ( x, y) xy ; 0 x, 0 y. Compute (i) P X 8 (ii) / Y PY / X (iii) P( X Y) and (iv) P ( X Y ). (8) 3. If the joint distribution function of X and Y is given by x y e e ; x 0, y 0 F x, y. [A/M -5] 0 ; otherwise (i) Find the marginal densities of X and Y (ii) Are X and Y independent? (iii) P X 3, Y (8) 4. Let X, Y and Z be uncorrelated random variables with zero means and standard deviation 5,,9 respectively. If U X Y and V Y Z, find the correlation between U and V. (8) 5. Two independent random variable X and Y are defined by 4 ax ; 0 x 4 by ; 0 y fx x and fy y 0 ; otherwise 0 ; otherwise (i) Show that U=X+Y and V=X-Y are uncorrelated (ii) Find Correlation Coefficient of X and Y. [A/M-3] (8)

4 The two lines of regression are 8x 0y 66 0 and 40x 8y 4 0.The Variance of x is 9. Find (i) the mean value of x and y (ii) correlation coefficient between x and y and (iii) variance of y. [A/M -5] (8) 6. T he lifetime of a certain brand of an electric bulb may be considered as a random variable with mean 00h and standard deviation 50h. Find the probability, using central limit theorem, that the average lifetime of 60 bulbs exceeds 50h. (8) 7. A distribution with unknown mean has variance equal to.5. Use Central limit theorem to find how large a sample should be taken from the distribution in order that the probability will be atleast 0.95 that the sample mean will be within 0.5 of the population mean. (8) x y 8. If X and Y are independent random variables with p.d.f e, x 0 ; e, y 0respectively. X Find the density function of U X Y and V X Y. Are U and V independent? [N/D-3] (8) 9. Let( XY, ) be a two- dimensional non-negative continuous random variable having the joint density f x, y ( x y ) xye x y 4 ; 0, 0 0 ; otherwise. Find the density function ofu x y. (8)

5 UNIT III : CLASSIFICATION OF RANDOM PROCESS PART B (6 MARKS). The process X t whose probability distribution under certain conditions is given by, n at, n,,... n at P X t n at, n=0 at Show that it is not stationary (or evolutionary). If [N/D -0] [N/D-] (8) Xt = A cost+bsin t, where A and B are two independent random variable with =E 0 ; E E E A B A B, and λ is a constant.prove that X t is a strict sense stationary process of order. [A/M -5] (8). If Xt = Ycost+Zsint for all t where Y and Z are independent binary random variables each of which assumes the values - and with probabilities 3 and 3 respectively, prove that X t is W.S.S. [A/M-3] [N/D-5] (8) 3. If X t is a WSS process with auto correlation function RXX and if Y t Xt a X t a. Show that R R R a-r a YY XX XX XX [N/D -5] (8) 4. A transition probability matrix of a markov chain ( Xn, n,,3,...) having three states, and 3 are P and the initial distribution P (0) [ ]. Find (i) probability of PX ( 3) and (ii) P( X3, X 3, X 3, X0 ) (8)

6 5. A man either drives a car or catches a train to go to office each day. The never goes in a row by train but if he drives one day then the next day he is just as likely to drive again as he is travel by train. Now suppose that on the first day of the week the man tossed a fair die and drove to work if and only if 6 appeared. Find the probability that (i) he takes a train on the 3 rd day and (ii) he drives to work in the long run. [N/D-5] (8) 6. Given that WSS random process { X ( t)} 0cos(00 t ) where is a uniformly distributed over (, ). Prove that the process {X (t)} is correlation ergodic. [N/D-0] (8) 7. The random binary transmission process {X (t)} is a WSS process with mean and autocorrelation function R XX ( ), where T is a Constant. Find the mean and variance of the time average T of {X (t)} over (0, T). Is {X (t)} mean ergodic? [A/M-0] (8) 8. 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 (i) The probability that exactly 3 messages will arrive during 0 second interval (ii) The probability that the number of message arrivals in an interval of duration 5 seconds is between 3 and 7. (8) 9. Let {X (t)} is a Gaussian random process with { Xt ( )} 0 and C ( t, t ) 6e t t. Find (i) X (0) 8 (ii) X(0) X(6) 4. [A/M -] (8) XX

7 UNIT IV : CORRELATION AND SPECTRAL DENSITIES PART B (6 MARKS). If X t and Yt are two random process with auto correlation respectively then prove that R R 0 R 0 R and R. (8) XY XX YY. Consider two random process X t = 3 cos t and Y t = cos where is a random variable uniformly distributed in (0,π). Prove that 0 0 R R R XY XX YY XX YY t. (8) 3. Xt and auto correlation function Y t are zero mean and stochastically independent random process having RXX e and YY cos (a) the auto correlation function of W t X t Y t (b) the cross correlation function R respectively. Find Wt and 4. Given the power spectral density of a continuous process as, Z t X t Y t Zt. (8) 9 S XX Find the mean square value of the process. [N/D-] (8) 5. The auto correlation of a wide stationary process is given by R XX = e. Determine the power spectral density function of the random telegraph signal. [N/D-4,5] (8) 6. The auto correlation function of the random binary transmission RXX, for T X t is given by T. Find the power spectrum of the 0, for > T X t. [A/M-0] [A/M-5] (8)

8 ; R XX. ; Find the power spectral density of the process. [N/D-] (8) 7. The auto correlation function of a random process is given by 8. The power spectral density of a zero mean WSS process ; o S 0 ; otherwise. Find R and show that X t and 9. If the cross power spectral density of X t and Yt is S XY X t is given by X t o are uncorrelated. [A/M -] ib a,, 0 where a and b are constants. [M/J -3] 0, elsewhere Find the cross correlation function. (8) 0. State and prove Weiner-Khintchine Theorem. [N/D-0] [A/M-,M/J-4] (8)

9 UNIT V : LINEAR SYSTEMS WITH RANDOM INPUTS PART B (6 MARKS). If the input to a time invariant stable linear system is a WSS process, then the output will also be a WSS process. [M/J-3] (8). If Xt is a band limited process such that XX 0, R 0 R R 0 S, prove that XX XX XX [A/M-0] (8) 3. If Yt is the output process where an input process impulse response ht, the power spectral density of YY X t is applied to an L.T.I system with Yt is given by, where S : p.s.d of X t and S H S XX XX H : system transfer. [M/J -3], [N/D-] (8) 4. Let Xt be the input voltage and function H RXX R R jl Y t be the output voltage with system transfer. Also Xt is a stationary process with 0 e. Find (i) EY (ii) S (iii) YY YY E X and R. (8) 5. Consider a system with system transfer function. An input signal with auto correlation i function m + m is fed as input to the system. Find the mean and mean square value of the output. [M/J ] (8) Rc 6. A linear system is described by the impulse response ht e u t process whose ACF is Rc t. Assume an input B.Find the mean and ACF of the output process. [A/M-] (8)

10 7. A random process X t is the input to linear system whose impulse response t = e ; t 0.If the auto correlation of the process is R XX ht = e, find the power spectral density of the output processy t. [A/M-3] [N/D-5] (8) 8. If Nt is a band limited white noise centered at a carrier frequency 0 such that N0, for < S NN = 0, elsewhere 0. Find the auto correlation of the N t. (8) N0 9. Consider the Gaussian white noise of zero mean and power spectral density applied to a low pass Rc filter where transfer function is H f. j f Rc Find (i) the output spectral density (ii) Auto correlation function of the output process. (8) 0. If Y t Acos 0 N t distribution in (-π,π) and density ; where A is a constant,θ is a random variable with uniform S NN Assume that N 0 N t is a band limited Gaussian white noise with power spectral ; 0. Find the power spectral density 0 ; for elsewhere Y t. N t and θ are independent. [A/M-] (8)

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

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