MA2261 PROBABILITY AND RANDOM PROCESSES - …

1 MA2261 PROBABILITY AND RANDOM , , ,Assistant Professor of MathematicsDhanalakshmi College of EngineeringMobile: 9841168917 Website: - PROBABILITY AND RANDOM RANDOM VARIABLESD iscrete and continuous RANDOM variables Moments - Moment generating functions and their properties. Binomial, Poisson ,Geometric, Uniform, Exponential, Gamma and normal distributions Function of RANDOM TWO DIMENSIONAL RANDOM VARIBLESJ oint distributions - Marginal and conditional distributions Covariance - Correlation and Regression - Transformation of RANDOM variables - Central limit theorem (for iid RANDOM variables)

2 CLASSIFICATION OF RANDOM PROCESSESD efinition and examples - first order, second order, strictly stationary, wide-sense stationary and ergodic PROCESSES - Markov process - Binomial, Poisson and Normal PROCESSES - Sine wave process RANDOM telegraph CORRELATION AND SPECTRAL DENSITIESAuto correlation - Cross correlation - Properties Power spectral density Cross spectral density -Properties Wiener-Khintchine relation Relationship between cross power spectrum and cross correlation LINEAR SYSTEMS WITH RANDOM INPUTSL inear time invariant system - System transfer function Linear systems with RANDOM inputs Auto correlation and cross correlation functions of input and output white Book1.

3 Oliver C. Ibe, Fundamentals of Applied PROBABILITY and RANDOM PROCESSES , Elsevier, First Indian Reprint ( 2007) (For units 1 and 2) Jr. , PROBABILITY RANDOM Variables and RANDOM Signal Principles , Tata McGraw-Hill Publishers, Fourth Edition, New Delhi, 2002. (For units 3, 4 and 5).References1. Miller, and Childers, , PROBABILITY and RANDOM PROCESSES with applications to Signal Processing and Communications , Elsevier Inc., First Indian Reprint H. Stark and Woods, PROBABILITY and RANDOM PROCESSES with Applications to Signal Processing , Pearson Education (Asia), 3rd Edition, Hwei Hsu, Schaum s Outline of Theory and Problems of PROBABILITY , RANDOM Variables and RANDOM PROCESSES , Tata McGraw-Hill edition, New Delhi, Leon-Garcia,A, PROBABILITY and RANDOM PROCESSES for Electrical Engineering , Pearson Education Asia, Second Edition, Yates and Goodman, PROBABILITY and Stochastic PROCESSES , John Wiley and Sons, Second edition, Mathematics 2013 Prepared by , , , (Ph:9841168917) Page 1 SUBJECT NAME.

4 PROBABILITY & RANDOM Process SUBJECT CODE : MA 2261 MATERIAL NAME : University Questions MATERIAL CODE : JM08AM1004 Name of the Student: Branch: Unit I ( RANDOM Variables) Problems on Discrete & Continuous 1. A RANDOM variable Xhas the following PROBABILITY distribution. X 0 1 2 3 4 5 6 7 P(x) 0 k 2k 2k 3k 2k 22k 27kk Find: (1) The value of k (2) ( /2)PXX and (3) The smallest value of nfor which 1()2P Xn . (N/D 2010),(M/J 2012) 2. The PROBABILITY mass function of RANDOM variable X is defined as 2(0) 3P XC , 2(1) 410P XCC , (2)51P XC , where 0C and ()0P Xr if 0,1, 2r.

5 Find (1) The value of C (2) (02 /0)PXx (3) The distribution function of X (4) The largest value of X for which1()2Fx . (A/M 2010) 3. The PROBABILITY density function of a RANDOM variable X is given by Engineering Mathematics 2013 Prepared by , , , (Ph:9841168917) Page 2 , 01( )(2), 120, otherwiseXxxf xkxx . (1) Find the value of k . (2) Find ( )Px (3) What is /1 Pxx (4) Find the distribution function of()fx. (A/M 2011) 4. A continuous the 2, ()10, elsewherekxfxx . Find (1) the value of k (2) Distribution function ofX (3) (0)PX (N/D 2011) 5.

6 Show that for the PROBABILITY function 1, 1, 2, ( )()0, otherwisexxxp xP Xx ()EX does not exist. (N/D 2012) 6. The PROBABILITY function of an infinite discrete distribution is given by 1() (1, 2, 3, ..)2jP Xjj Find (1) Mean of X (2) ( is even)PX and (3) ( is divisible by 3)PX (N/D 2011) Moments and Moment Generating Function 1. Find the MGF of the two parameter exponential distribution whose density function is given by ()( ), xaf xex a and hence find the mean and variance. (A/M 2010) Engineering Mathematics 2013 Prepared by , , , (Ph:9841168917) Page 3 2.

7 Derive the of Poisson distribution and hence or otherwise deduce its mean and variance. (A/M 2011) 3. If the PROBABILITY density of X is given by 2(1) for 01()0, otherwisexxfx , find its rth moment. Hence evaluate 221EX . (N/D 2012) 4. Find the of the RANDOM variableXhaving the PROBABILITY density function 2, 0()40, elsewherexxexfx . Also deduce the first four moments about the origin. (N/D 2010),(M/J 2012) 5. Find MGF corresponding to the distribution 21, 0()20, otherwiseef and hence find its mean and variance. (N/D 2012) Problems on distributions 1.

8 If the PROBABILITY that an applicant for a driver s license will pass the road test on any given trial is What is the PROBABILITY that he will finally pass the test (1) On the fourth trial and (2) In less than 4 trials? (A/M 2010) 2. 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 2010) 3. 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.

9 If 3 students are selected at RANDOM from this set, what is the PROBABILITY that exactly 2of them will have marks over 70? (A/M 2011) 4. Assume that the reduction of a person s oxygen consumption during a period of Transcendental Meditation ( ) is a continuous RANDOM variable X normally distributed with mean cc/mm and cc/min. Determine the PROBABILITY that during a period of a person s oxygen consumption will be reduced by Engineering Mathematics 2013 Prepared by , , , (Ph:9841168917) Page 4 (1) at least cc/min (2) at most cc/min (3) anywhere from to cc/mm. (N/D 2012) 5.

10 LetXand Ybe independent normal variates with mean 45 and 44 and standard deviation 2 and respectively. What is the PROBABILITY that randomly chosen values ofXand Ydiffer by or more? (N/D 2011) 6. Given that Xis distributed normally, if (45) and (64) , find the mean and standard deviation of the distribution. (M/J 2012) 7. If X and Yare independent RANDOM variables following (8, 2)N and 12, 4 3N respectively, find the value of such that 222P X YP XY . (N/D 2010) 8. The time in hours required to repair a machine is exponentially distributed with parameter 1 / 2.