SUBJECT NAME : Probability & Random Processes SUBJECT CODE : MA645 MATERIAL NAME : Additional Problems MATERIAL CODE : JM08AM004 REGULATION : R03 UPDATED ON : March 05 (Scan the above QR code for the direct download of this material) Name of the Student: Branch: Unit I (Random Variables) Problems on Discrete & Continuous RVs ) A random variable X has the following probability function: X 0 3 4 5 6 7 P(X) 0 K K K 3K K K 7K +K a) Find K b) Evaluate P X 6, P X 6 c) Find P X, P X 3, P X 5 ) Suppose that X is a continuous random variable whose probability density function is given C 4x x, 0 x f( x) 0, otherwise by 3) A random variable X has the pdf (a) findc (b) find P X x, 0 x f( x) Find (i) 0, otherwise 3 P X 4 (iii) 3 P X / X 4 (iv) 3 P X / X 4 4) If a random variable X has the pdf (b) P X (c) PX 3 5, x f( x) 4 0, otherwise P X (ii) Find (a) P X Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page
5) The amount of time, in hours that a computer functions before breaking down is a continuous random variable with probability density function given by x 00 f( x) e, x 0 What is the probability that (a) a computer will function between 0, x 0 50 and 50 hrs before breaking down (b) it will function less than 500 hrs 6) A random variable X has the probability density function Find, c d f, P X 5, P X 7 7) If the random variable X takes the values,,3 and 4 such that x xe, x 0 f( x) 0, otherwise P X 3P X P X 3 5P X 4 Find the probability distribution x F( x) x e ; x 0 8) The distribution function of a random variable X is given by Find the density function, mean and variance of X 9) A continuous random variable X has the distribution function 0, x 4 F( x) k( x ), x 3 Find k, probability density function f( x ), P X 0, x 30 0) A test engineer discovered that the cumulative distribution function of the lifetime of an x 5 equipment in years is given by F( x) e, x 0 0, x 0 i) What is the expected life time of the equipment? ii) What is the variance of the life time of the equipment? Moments and Moment Generating Function ) Find the moment generating function of RV X whose probability function P( X x), x,, Hence find its mean and variance x ) The density function of random variable X is given by f ( x) Kx( x), 0 x Find K, mean, variance and rth moment x 3 e, x 0 3) Let X be a RV with pdf f( x) 3 Find the following 0, Otherwise a) P(X > 3) b) Moment generating function of X c) E(X) and Var(X) Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page
x, 0 x 4) Find the MGF of a RV X having the density function f( x) Using the 0, otherwise generating function find the first four moments about the origin 5) Define Geometric distribution and find the MGF, Mean and Variance of the Geometric distribution 6) Write the pdf of Uniform distribution and find the MGF, Mean and Variance 7) Define Exponential distribution and find the MGF, Mean and Variance of the Exponential distribution 8) Define Normal distribution and find the MGF, Mean and Variance of the Normal distribution Problems on distributions ) The mean of a Binomial distribution is 0 and standard deviation is 4 Determine the parameters of the distribution ) If 0% of the screws produced by an automatic machine are defective, find the probability that of 0 screws selected at random, there are (i) exactly two defectives (ii) atmost three defectives (iii) atleast two defectives and (iv) between one and three defectives (inclusive) 3) In a certain factory turning razar blades there is a small chance of /500 for any blade to be defective The blades are in packets of 0 Use Poisson distribution to calculate the approximate number of packets containing (i) no defective (ii) one defective (iii) two defective blades respectively in a consignment of 0,000 packets 4) The number of monthly breakdown of a computer is a random variable having a Poisson distribution with mean equally to 8 Find the probability that this computer will function for a month a) Without a breakdown b) With only one breakdown and c) With atleast one breakdown t 8 5) If the mgf of a random variable X is of the form (04e 06), what is the mgf of 3X Evaluate E X 6) A discrete RV X has moment generating function Var X and P X 7) If X is a binomially distributed RV with EX ( ) and 5 t Find 3 M X () t e 4 4 E X, 4 Var( X ), find P X 5 3 8) If X is a Poisson variate such that P X 9P X 4 90P X 6, find the mean and variance 9) The number of personal computer (PC) sold daily at a CompuWorld is uniformly distributed with a minimum of 000 PC and a maximum of 5000 PC Find the following Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page 3
(i) The probability that daily sales will fall between,500 PC and 3,000 PC (ii) What is the probability that the CompuWorld will sell at least 4,000 PC s? (iii) What is the probability that the CompuWorld will exactly sell,500 PC s? 0) Suppose that a trainee soldier shoots a target in an independent fashion If the probability that the target is shot on any one shot is 08 (i) What is the probability that the target would be hit on 6 th attempt? (ii) What is the probability that it takes him less than 5 shots? (iii) What is the probability that it takes him an even number of shots? ) A die is cast until 6 appears What is the probability that it must be cast more than 5 times? ) The length of time (in minutes) that a certain lady speaks on the telephone is found to be random phenomenon, with a probability function specified by the function x 5 f( x) Ae, x 0 (i) Find the value of A that makes f(x) a probability density 0, otherwise function (ii) What is the probability that the number of minutes that she will talk over the phone is (a) more than 0 minutes (b) less than 5 minutes and (c) between 5 and 0 minutes 3) If the number of kilometers that a car can run before its battery wears out is exponentially distributed with an average value of 0,000 km and if the owner desires to take a 5000 km trip, what is the probability that he will be able to complete his trip without having to replace the car battery? Assume that the car has been used for same time 4) The mileage which car owners get with a certain kind of radial tyre is a random variable having an exponential distribution with mean 40,000 km Find the probabilities that one of these tyres will last (i) atleast 0,000 km and (ii) atmost 30,000 km 5) If a continuous random variable X follows uniform distribution in the interval 0, and a continuous random variable Y follows exponential distribution with parameter, find such that P X P Y 6) If X is exponentially distributed with parameter, find the value of K there exists P X k P X k a 7) State and prove memoryless property of Geometric distribution 8) State and prove memoryless property of Exponential distribution 9) The weekly wages of 000 workmen are normall distributed around a mean of Rs70 with a SD of Rs5 Estimate the number of workers whose weekly wages will be (i) between Rs 69 and Rs 7, (ii) less than Rs 69 and (iii) more than Rs 7 0) In a test on 000 electric bulbs, it was found that the life of a particular make, was normally distributed with an average life of 040 hours and SD of 60 hours Estimate the number of bulbs likely to burn for (i) more than 50 hours, (ii) less than 950 hours and (iii) more than 90 hours but less than 60 hours Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page 4
Function of random variable x, x 5 ) Let X be a continuous random variable with pdf f( x), find the 0, otherwise probability density function of X 3 ) If X is a uniformly distributed RV in,, find the pdf of tan Y X 3) If X has an exponential distribution with parameter, find the pdf of Y X x 4) If the pdf of X is f ( x) e, x 0, find the pdf of Y X 5) If X is uniformly distributed in0, find the pdf of Y X Unit II (Two Dimensional Random Variables) Joint distributions Marginal & Conditional ) The two dimensional random variable (X,Y) has the joint density function x y f ( x, y), x 0,,; y 0,, Find the marginal distribution of X and Y and 7 the conditional distribution of Y given X = x Also find the conditional distribution of X given Y = ) The joint probability mass function of (X,Y) is given by P( x, y) K x 3 y, x 0,, ; y,, 3 Find all the marginal and conditional probability distributions Also find the probability distribution of X Y and P X Y 3 3) If the joint pdf of a two dimensional random variable (X,Y) is given by K(6 x y),0 x, y 4 f ( x, y) Find the following (i) the value of K; (ii) 0,otherwise P x, y 3 ; (iii) P x y 3 ; (iv) P x / y 3 4) If the joint pdf of a two dimensional random variable (X,Y) is given by xy x,0 x, 0 y f ( x, y) 3 Find (i) 0,otherwise P X ; (ii) P Y X P Y / X Check whether the conditional density functions are valid ; (iii) Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page 5
5) The joint pdf of the random variable (X,Y) is given by x y f ( x, y) Kxye, 0 x, y Find the value of K and Prove that X and Y are independent 6) If the joint distribution function of X and Y is given by x y F( x, y) e e, x 0, y 0 and "0" otherwise (i) Are X and Y independent? (ii) Find P X 3, Y Covariance, Correlation and Regression ) Define correlation and explain varies type with example ) Find the coefficient of correlation between industrial production and export using the following data: Production (X) 55 56 58 59 60 60 Export (Y) 35 38 37 39 44 43 6 44 3) Let X and Y be discrete random variables with probability function x y f ( x, y), x,,3; y, Find (i) Cov X, Y (ii) Correlation co efficient 4) Let X and Y be random variables having joint density function 3 x y, 0 x, y f ( x, y) Find the correlation coefficient ( XY, ) 0, otherwise 5) Let X,Y and Z be uncorrelated random variables with zero means and standard deviations 5, and 9 respectively If U X Y and V Y Z, find the correlation coefficient between U and V 6) 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 7) From the data, find (i) The two regression equations (ii) The coefficient of correlation between the marks in Economics and Statistics (iii) The most likely marks in statistics when a mark in Economics is 30 Marks in Economics 5 8 35 3 3 36 9 Marks in Statistics 43 46 49 4 36 3 3 38 34 3 30 33 39 Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page 6
8) The two lines of regression are 8x 0y + 66 = 0, 40x 8y 4 = 0 The variance of X is 9 Find (i) the mean values of X and Y (ii) correlation coefficient between X and Y (iii) Variance of Y 9) The joint pdf of a two dimensional random variable is given by f ( x, y) ( x y); 0 x, 0 y Find the following 3 (i) The correlation co efficient (ii) The equation of the two lines of regression (iii) The two regression curves for mean Transformation of the random variables ) If X is a uniformly distributed RV in,, find the pdf of tan Y X ) Let (X,Y) be a two dimensional non negative continuous random variables having the x y 4, 0, 0 joint probability density function f ( x, y) xye x y Find the density 0, elsewhere function of U X Y 3) If X and Y are independent exponential random variables each with parameter, find the pdf of U = X Y 4) Let X and Y be independent random variables both uniformly distributed on (0,) Calculate the probability density of X + Y 5) Let X and Y are positive independent random variable with the identical probability density x function f ( x) e, x 0 Find the joint probability density function of U X Y and X V Are U and V independent? Y 6) If the joint probability density of X and X is given by find the probability of Y X X X xx e x x, 0, 0 f ( x, x), 0, elsewhere x, 0 x 7) If X is any continuous RV having the pdf f( x), andy e X, find the 0, otherwise pdf of the RV Y 8) If the joint pdf of the RVs X and Y is given by X the RV U Y, 0 x y f ( x, y) find the pdf of 0, otherwise Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page 7
9) Let X be a continuous random variable with pdf probability density function of X 3 x, x 5 f( x), find the 0, otherwise Unit III (Random Processes) Verification of SSS and WSS process ) Classify the random process and give example to each ) Let X Acos( n) Bsin( n) where A and B are uncorrelated random variables with n E B0 andvar A Var B E A Show that X n is covariance stationary 3) A stochastic process is described by X( t) Asin t Bcos t where A and B are independent random variables with zero means and equal standard deviations show that the process is stationary of the second order 4) If X( t) Y cost Z sint, where Y and Z are two independent random variables with and is a constants Prove that Xt () E( Y ) E( Z) 0, E( Y ) E( Z ) strict sense stationary process of order (WSS) 5) At the receiver of an AM radio, the received signal contains a cosine carrier signal at the carrier frequency 0 with a random phase that is uniformly distributed over 0, The received carrier signal is X( t) Acos t order stationary Problems on Markov Chain ) Consider a Markov chain n ; 09 0 probability matrix P 0 08 0 is a Show that the process is second X n with state space S, i) Is chain irreducible? ii) Find the mean recurrence time of states and iii) Find the invariant probabilities and one step transition ) A raining process is considered as two state Markov chain If it rains, it is considered to be state 0 and if it does not rain, the chain is in state The transitions probability of the Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page 8
06 04 Markov chain is defined as P Find the probability that it will rain for 3 days 0 08 Assume the initial probabilities of state 0 and state as 04 and 06 respectively 3) A person owning a scooter has the option to switch over to scooter, bike or a car next time with the probability of (03, 05, 0) If the transition probability matrix is 04 03 03 0 05 03 What are the probabilities vehicles related to his fourth purchase? 05 05 05 4) Assume that a computer system is in any one of the three states: busy, idle and under repair respectively denoted by 0,, Observing its state at pm each day, we get the transition 06 0 0 probability matrix as P 0 08 0 Find out the 3 rd step transition probability 06 0 04 matrix Determine the limiting probabilities 5) Two boys B and B and two girls G and G are throwing a ball from one to the other Each boys throws the ball to the other boy with probability / and to each girl with probability /4 On the other hand each girl throws the ball to each boy with probability / and never to the other girl In the long run, how often does each receive the ball? 6) A housewife buys 3 kinds of cereals A, B, C She never buys the same cereal in successive weeks If she buys cereal A, the next week she buys cereal B However if she buys B or C the next week she is 3 times as likely to buy A as the other cereal How often she buys each of the 3 cereals? 7) Three boys A, B, C are throwing a ball each other A always throws the ball to B and B always throws the ball to C, but C is just as likely to throw the ball to B as to A Find the transition matrix and classify the states 8) The tpm of a Markov chain with three states 0,, is 3 / 4 / 4 0 P / 4 / / 4 and the 0 3 / 4 / 4 initial state distribution of the chain is P X0 i / 3, i 0,, Find (i) P X and (ii) P X, X, X, X 3 0 Poisson process ) Prove that the Poisson process is Covariance stationary ) 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 mins (i) Exactly 4 customers arrive and (ii) More than 4 customers arrive Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page 9
3) If customers arrive at a counter in accordance with a Poisson process with a mean rate of 3 per minute, find the probability that the interval between consecutive arrivals is (i) more than minute (ii) between minute and minutes (iii) 4 minutes or less 4) A radar emits particles at the rate of 5 per minute according to Poisson distribution Each particles emitted has probability 06 Find the probability that 0 particles are emitted in a 4 minutes period 5) Queries presented in a computer data base are following a Poisson process of rate 6 queries per minute An experiment consists of monitoring the data base for m minutes and recording Nm ( ) the number of queries presented i) What is the probability that no queries in a one minute interval? ii) What is the probability that exactly 6 queries arriving in one minute interval? iii) What is the probability of less than 3 queries arriving in a half minute interval? Unit IV (Correlation and Spectral densities) Section I ) Determine the mean and variance of process given that the auto correlation function 4 R 5 6 ) A stationary random process has an auto correlation function and is given by R 3) If () 5 36 65 4 Xt and () Find the mean and variance of the process Yt are two random processes then R ( ) R (0) R (0) where R ( ) and RYY ( ) are their respective auto correlation function 4) If Xt () and Yt () are two random processes then XY YY RXY ( ) R (0) RYY (0) where R ( ) and RYY ( ) are their respective auto correlation function Section II 5) The auto correlation of a stationary random process is given by ( ) b R ae, b 0 Find the spectral density function 6) The auto correlation of the random binary transmission is given by R, ( ) T 0, for T Find the power spectrum for T Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page 0
Note: By putting T =, the above problem can be ask R, for ( ) 0, for 7) Show that the power spectrum of the auto correlation function e Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page 3 4 is 8) Find the power spectral density of a WSS process with auto correlation function ( ) R, 0 e 9) Find the power spectral density of the random process, if its auto correlation function is given by R ( ) e cos 0) Find the power spectral density function whose auto correlation function is given by A R ( ) cos( 0 ) Section III ) If the power spectral density of a WSS process is given by S find the auto correlation function of the process b a ( ) a 0, ) The power spectral density of a zero mean WSS process Xt () is given by S,, a ( ) Find R ( ) and show that Xt () and Xt 0, elsewhere a are uncorrelated a, a 3) Find the autocorrelation function of the process Xt (), for which the spectral density is, given by S( ) 0, 4) The cross power spectrum of real random processes Xt () and Yt () S XY a jb, ( ) Find the cross correlation function 0, elsewhere Section IV 5) If Y( t) X( t a) X( t a),prove that R ( ) R ( ) R ( a) R ( t a) Hence prove that YY S a S YY ( ) 4sin ( ) ( ) Xt and Yt () 6) () is given by are zero mean and stochastically independent random process having autocorrelation function R ( ) e, R ( ) cos respectively Find (i) the auto YY
correlation function of W( t) X( t) Y( t) and Z( t) X( t) Y( t) (ii) The cross correlation function of W() t and Zt () 7) If Xt () and Yt () are independent with zero means Find the auto correlation function of Zt () where Z( t) a bx( t) cy ( t) and Y( t) cos t are two random processes where is a random variable uniformly distributed in 0, Prove that 8) If X( t) 3cos t R 0 RYY 0 RXY 9) Two random process Xt () and Yt () are given by X( t) Acos t ; Y( t) Asint where A and are constants and " " is a uniform random variable over 0 to Find the cross correlation function 0) If () Xt is a process with mean ( t) 3 and auto correlation R t, t 9 4e Determine the mean, variance of the random variable Z X(5) and W X(8) 0 Unit V (Linear systems with Random inputs) ) Prove that if the input Xt () is WSS then the output Yt () is also WSS ) If Xt () is the input voltage to a circuit and Yt () is the output voltage, stationary random process with x 0 if the system function is given by H( ) 3) If () Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page () Xt is a and ( ) R e Find y, S ( ) and S ( ), i Xt is a band limited process such that S ( ) 0,, prove that R (0) R ( ) R (0) 4) Let Xt () be a random process which is given as input to a system with the system transfer function H( ), 0 0 If the autocorrelation function of the input process is N 0 ( ), find the auto correlation of the output process 5) If 0 uniform distribution in, and Nt () Y( t) Acos t N( t) where A is a constant, is a random variable with a is a band limited Gaussian white noise with a N0 power spectral density SNN ( ) for 0 B and SNN ( ) 0,elsewhere Find the power spectral density of Yt (), assuming that Nt () and are independent YY
N 0 6) Consider a white Gaussian noise of zero mean and power spectral density low pass RC filter whose transfer function is H( f) i frc Find the autocorrelation function of the output random process 7) A WSS random process Xt () with auto correlation R ( ) Ae applied to a where A and are real positive constants, is applied to the input of an linear time invariant (LTI) system with bt impulse response h( t) e u( t) where b is a real positive constant Find the auto correlation of the output Yt () of the system t 8) An linear time invariant (LIT) system has an impulse response h( t) e u( t) Find the output auto correlation function RYY ( ) corresponding to an input Xt () ----All the Best---- Prepared by CGanesan, MSc, MPhil, (Ph:9846897) Page 3