Noida Institute of Engineering and Technology Statistics & Probability BAS0303 PDF

Noida Institute of Engineering and Technology, Greater Noida Statistics & Probability BAS0303 Unit: III Probability distribution Dr. Anil Agarwal Associate Professor B.Tech-3r...

Noida Institute of Engineering and Technology, Greater Noida Statistics & Probability BAS0303 Unit: III Probability distribution Dr. Anil Agarwal Associate Professor B.Tech-3rd Sem Dept. of (DS/AIML/AI) Mathematics Dr. Anil Agarwal Unit III 1 12/13/2024  Sequence of Content (1)Name of Subject with code, Course and Subject teacher. (2)Brief Introduction of Faculty. (3)Evaluation Scheme. (4)Subject Syllabus. (5)Branch Wise Application. (6)Course Objective. (7)Course Outcomes(COs). (8)Program Outcomes(POs). (9)COs and POs Mapping. (10)Program Specific Outcomes(PSOs) (11) COs and PSOs Mapping. (12)Program Educational Objectives(PEOs). 12/13/2024 Dr. Anil Agarwal Unit-III 2  Sequence of Content (13)Result Analysis. (14) End Semester Question Paper Templates. (15) Prequisite /Recap. (16) Brief Introduction about the Subject. (17) Unit Content. (18) Unit Objective. (19) Topic Objective/Topic Outcome. (20) Lecture related to topic. (21) Daily Quiz. (22) Weekly Assignment. (23) Topic Links. 12/13/2024 Dr. Anil Agarwal Unit-III 3  Sequence of Content (24) MCQ(End of Unit). (25) Glossary questions. (26) Old question Papers(Sessional + University). (27) Expected Questions For External Examination. (28) Recap of Unit. 12/13/2024 Dr. Anil Agarwal Unit-III 4  Faculty Introduction Name : Dr. Anil Agarwal Designation: Associate professor Department: Mathematics Teaching Experience: 22years Ph.D. : Agra University, Agra. 12/13/2024 Unit-III 5  Evaluation Scheme Sl. Subject Periods Evaluation Scheme End No. Codes Semeste Credi Subject Name Total t r L T P CT TA TOTAL PS TE PE WEEKS COMPULSORY INDUCTION PROGRAM 1 BAS0303 Statistics and 3 1 0 30 20 50 100 150 4 Probability 2 ACSE030 Discrete Structures 3 0 0 30 20 50 100 150 3 6 3 ACSE030 Computer Organization 3 0 0 30 20 50 100 150 3 & 5 Architecture 4 ACSE030 Object Oriented 3 0 0 30 20 50 100 150 3 Techniques 2 using Java 5 ACSE030 Data Structures 3 1 0 30 20 50 100 150 4 1 6 ACSAI030 Introduction to Artificial 3 0 0 30 20 50 100 150 3 Intelligence 1 7 ACSE035 Object Oriented 0 0 2 25 25 50 1 Techniques 2 using Java Lab 8 ACSE035 Data Structures Lab 0 0 2 25 25 50 1 1 9 ACSAI035 Introduction to Artificial 0 0 2 25 25 50 1 Intelligence Lab 1 10 ACSE035 Internship Assessment-I 0 0 2 50 50 1 9 ANC030 Cyber Security*/ 11 12/13/2024 Environmental Dr. Anil Agarwal Unit-III 6 1/ Science * (Non 2 0 0 30 20 50 50 100 0  Syllabus of BAS0303 UNIT-I Descriptive measures 8 Hours Measures of central tendency – mean, median, mode, measures of dispersion – mean deviation, standard deviation, quartile deviation, variance, Moment, Skewness and kurtosis, least squares principles of curve fitting, Covariance, Correlation and Regression analysis, Correlation coefficient: Karl Pearson coefficient, rank correlation coefficient, uni-variate and multivariate linear regression, application of regression analysis, Logistic Regression, time series analysis- Trend analysis (Least square method). UNIT-II Probability and Random variable 8 Hours Probability Definition, The Law of Addition, Multiplication and Conditional Probability, Bayes’ Theorem, Random variables: discrete and continuous, probability mass function, density function, distribution function, Mathematical expectation, mean, variance. Moment generating function, characteristic function, Two dimensional random variables: probability mass function, density function, UNIT-III Probability distribution 8 Hours Probability Distribution (Continuous and discrete- Normal, Exponential, Binomial, Poisson distribution), Central Limit theorem UNIT-IV Test of Hypothesis & Statistical Inference 8 Hours Sampling and population, uni-variate and bi-variate sampling, re-sampling, errors in sampling, Sampling distributions, Hypothesis testing- p value, z test, t test (For mean), Confidence intervals, F test; Chi-square test, ANOVA: One way ANOVA, Statistical Inference, Parameter estimation, Least square estimation method, 12/13/2024 Maximum Likelihood estimation. 7  Branch wise Application Data Analysis Artificial intelligence Digital Communication: Information theory and coding. Dr. Anil Agarwal Unit III 12/13/2024 8  Course Objective The objective of this course is to familiarize the engineers with concept of Statistical techniques, probability distribution, hypothesis testing and ANOVA and numerical aptitude. It aims to show case the students with standard concepts and tools from B. Tech to deal with advanced level of mathematics and applications that would be essential for their disciplines. The student will be able to understand: The concept of Descriptive measurements. The concept of probability & Random variable. Probability distributions. The concept of hypothesis testing & Statistical inferences. The concept of numerical aptitude. Dr. Anil Agarwal Unit III 12/13/2024 9  Course Outcome CO1: Understand the concept of moments, skewness, kurtosis, correlation, curve fitting and regression analysis, Time-Series analysis etc. CO2: Understand the concept of Probability and Random variables. CO3: Remember the concept of probability to evaluate probability distributions. CO4: Apply the concept of hypothesis testing and estimation of parameters. CO5: Solve the problems of Time & Work, Pipe & Cistern, Time, Speed & Distance, Boat & Stream, Sitting arrangement , Clock & Calendar etc. Dr. Anil Agarwal Unit II 12/13/2024 10  PO S.No Program Outcomes (POs) PO 1 Engineering Knowledge PO 2 Problem Analysis PO 3 Design/Development of Solutions Conduct Investigations of Complex PO 4 Problems PO 5 Modern Tool Usage PO 6 The Engineer & Society PO 7 Environment and Sustainability PO 8 Ethics PO 9 Individual & Team Work PO 10 Communication PO 11 Project Management & Finance PO 12 Lifelong Learning 12/13/2024 Dr. Anil Agarwal Unit-III 11  CO-PO Mapping(CO3) Sr. Cours PO PO PO PO PO PO PO PO PO PO PO PO No e 1 2 3 4 5 6 7 8 9 10 11 12 Outco me 1 CO 1 3 3 3 3 1 1 2 2 CO 2 3 3 3 2 1 1 2 2 3 CO 3 3 2 3 2 1 1 1 4 CO 4 3 2 2 3 1 1 1 5 CO.5 3 3 2 2 1 1 1 2 2 *1= Low *2= Medium *3= High 12/13/2024 Dr. Anil Agarwal Unit-III 12  PSO 12/13/2024 Dr. Anil Agarwal Unit-III 13  CO-PSO Mapping(CO3) CO PSO 1 PSO 2 PSO 3 CO1 3 2 1 CO2 1 2 1 CO3 2 2 2 CO4 3 2 1 CO5 3 2 2 *1= Low *2= Medium *3= High 12/13/2024 Dr. Anil Agarwal Unit-III 14  Program Educational Objectives(PEOs) PEO-1: To have an excellent scientific and engineering breadth so as to comprehend, analyze, design and provide sustainable solutions for real-life problems using state-of-the-art technologies. PEO-2: To have a successful career in industries, to pursue higher studies or to support entrepreneurial endeavors and to face the global challenges. PEO-3: To have an effective communication skills, professional attitude, ethical values and a desire to learn specific knowledge in emerging trends, technologies for research, innovation and product development and contribution to society. PEO-4: To have life-long learning for up-skilling and re-skilling for successful professional career as engineer, scientist, entrepreneur and bureaucrat for betterment of society. Dr. Anil Agarwal Unit-III 12/13/2024 15  Result Analysis Branc Semes Sectio No. of No. % h ter ns enroll Passe Passe ed d d Stude Stude nts nts AIML III A, B, C 199 199 100% Dr. Anil Agarwal Unit-III 12/13/2024 16  End Semester Question Paper Template C:\Users\om\ Desktop\Downloads\100 Marks Q Dr. Anil Agarwal Unit-III 12/13/2024 17  Prerequisite and Recap(CO3)  Knowledge of Maths -I of B.Tech.  Knowledge of Maths -II of B.Tech.  Knowledge of Basic Statistics. 12/13/2024 Dr. Anil Agarwal Unit-III 18  Brief Introduction about the subject In first four modules, we will discuss Statistics and probability. In 5th module we will discuss aptitude part. 12/13/2024 Dr. Anil Agarwal Unit-III 19  Unit Content(CO3) Introduction of Probability distributions Binomial Distribution Poisson Distribution Normal Distribution Exponential Distribution 12/13/2024 20 Dr. Anil Agarwal Unit-III  Unit Objective(CO3) 1. A basic knowledge in probability theory. 2. The student is able to reflect developed mathematical methods in probability and statistics. 3. Understand the concept of Probability and its usage in various business applications. 4. The binomial distribution model allows us to compute the probability of observing a specified number of "successes" when the process is repeated a specific number of times 5. Poisson Distribution is a tool that helps to predict the probability of certain events from happening when you know how often the event has occurred. 6. To learn the characteristics of a typical normal curve. 7. To explore the key properties, such as the moment- generating 12/13/2024 function,Dr. Anil mean Agarwal and Unit-III variance, 21 of  Topic Objective (CO3) The probability distributions are very much helpful for making predictions. Estimates and predictions form an important part of research investigation. With the help of Probability distributions, we make estimates and predictions for the further analysis. 12/13/2024 Dr. Anil Agarwal Unit-III 22  Probability distributions(CO3) Theoretical probability distribution Continuou Discrete s probability probabilit distribution y s distributio ns Binomial Normal Distribution Distribution Poisson t- Distribution Distribution F-Distribution 12/13/2024 Dr. Anil Agarwal Unit-III 23  Binomial Distribution(CO3) Binomial Probability Distribution: Probability distribution defined as follows is known as binomial Probability distribution. , Where n is no of trial which are finite ,r be the success in n trials and , p is probability of success and q is probability of failure. Assumptions For Binomial distribution: n, the number of trials is finite Each trial has only two possible outcomes usually called success and failure. All trials are independent. p and q is constant for all trials. 12/13/2024 Dr. Anil Agarwal Unit-III 24  Cont…(CO3) Recurrence or recursion formula: ….(1) Equation (1) denote binomial distribution. ….(2) By equation (1) and (2) …..(3) Equation (3) is known as Recurrence Formula. 12/13/2024 Dr. Anil Agarwal Unit-III 25  Cont…(CO3) Mean Of Binomial distribution: For Binomial distribution By expanding we have Hence mean of binomial distribution is np. 12/13/2024 Dr. Anil Agarwal Unit-III 26  Cont…(CO3) Variance of Binomial Distribution: 12/13/2024 Dr. Anil Agarwal Unit-III 27  Cont…(CO3) Hence the Variance of binomial distribution is and Standard deviation is. Moment generating function of binomial Distribution: i. About origin ii. About mean 12/13/2024 Dr. Anil Agarwal Unit-III 28  Cont…(CO3) Applications of Binomial Distribution: 1. In problem concerning no. of defectives in sample production line. 2. In estimation of reliability of systems. 3. No. of rounds fired from a gun hitting a target. 4. In radar detection. Q1. If 10% of bolts are produced by a machine are defective , determine the probability that out of 10 bolts chosen at random i. 1 ii. None iii. At most 2 bolts will be defective 12/13/2024 Dr. Anil Agarwal Unit-III 29  Problems based on Binomial Distribution(CO3) Solution: let p and q are the probability of defective and non defective bolts respectively. and n=10 (no of bolts chosen) The Probability of r defective bolts out of n bolt chosen at random is given by i. Here r=1, ii. Here r=0 12/13/2024 Dr. Anil Agarwal Unit-III 30  Problems based on Binomial distribution(CO3) iii.Prob.that at most 2 bolts will be defective =( =45 From(4).Required Probability= Q2. Out of 800 families with 4 children each, how many families would be expected to have i. 2 boys and 2 girls ii. At least one boy iii no girl iv. Atmost two girls? Assume equal probability for boys and girls. 12/13/2024 Dr. Anil Agarwal Unit-III 31  Cont…(CO3) Solution: Probability for boys and girls are equal n=4 N=800 i. The expected number of families having 2 boys and 2 girls ii. The expected number of families having at least one boy 12/13/2024 Dr. Anil Agarwal Unit-III 32  Cont…(CO3) iii. The expected number of families having no girl i.e. having 4 boys iv. The expected number of families having almost two girls i.e. having at least 2 boys 12/13/2024 Dr. Anil Agarwal Unit-III 33  Daily Quiz(CO3) 1. Four persons in a group of 20 are graduates. If 4 persons are selected at random from 20, find the probability that all 4 are graduates. Ans: 0.0016 2. The Prob. that a bulb produced by a factory will fuse after use of 150 days is 0.05. Find the probability that out of 5 such bulbs at least one bulb will fuse after use of 150 days of use. Ans: 12/13/2024 Dr. Anil Agarwal Unit-III 34  Poisson Distribution(CO3) Poisson distribution: Probability distribution defined as follows is known as Poisson Probability distribution. 𝑒 −𝜆 𝜆 𝑟 𝑃 ( 𝑋 =𝑟 ) = , ( 𝑟 = 0 , 1 , 2 , 3 …. ) 𝑟! Where Recurrence formula for Poisson Distribution: ……..(2) Poisson distribution 12/13/2024 Dr. Anil Agarwal Unit-III 35  Cont…(CO3) This is called the recurrence or recursion formula for Poisson distribution. Mean of the Poisson distribution: Mean 12/13/2024 Dr. Anil Agarwal Unit-III 36  Cont…(CO3) Mean for Poisson distribution. Variance of Poisson Distribution: 12/13/2024 Dr. Anil Agarwal Unit-III 37  Cont…(CO3) 12/13/2024 Dr. Anil Agarwal Unit-III 38  Cont…(CO3) Hence , the Variance of the Poisson distribution is also. Applications of Poisson Distribution: i. Arrival pattern of the defective vehicles in a workshop. ii. Patients in hospitals. iii. Telephone calls. iv. Emission of radioactive particles. 12/13/2024 Dr. Anil Agarwal Unit-III 39  Problem based on Poisson Distributions(CO3) Q1. If the Variance of the Poisson distribution is 2 , find the probability for r=1,2,3,4 from the recurrence relation of the Poisson Distribution. Also find Solution: Given that Variance Recurrence relation for Poisson distribution Poisson Distribution So Now putting r=0,1,2,3 in equation (1) 12/13/2024 Dr. Anil Agarwal Unit-III 40  Cont…(CO3) Now to calculate Q2. Fit a Poisson distribution to the following data and calculate theoretical frequencies. 12/13/2024 Dr. Anil Agarwal Unit-III 41  Cont…(CO3) Deaths: 0 1 2 3 4 Frequenci 122 60 15 2 1 Mean of es Poisson distribution Required Poisson distribution 12/13/2024 Dr. Anil Agarwal Unit-III 42  Cont…(CO3) r N.P(r) Theoretical frequencies 0 121 1 61 2 15 3 3 4 0 12/13/2024 Dr. Anil Agarwal Unit-III Total=200 43  Topic objective of Normal Distribution(CO3) To define the probability density function of a normal random variable. To learn the characteristics of a typical normal curve. To explore the key properties, such as the moment-generating function, mean and variance, of a normal random variable. 12/13/2024 Dr. Anil Agarwal Unit-III 44  Normal Distribution(CO3) Normal Distribution: The general equation of the normal distribution is given by ( ) 2 1 𝑥 −𝜇 1 − 2 𝜎 𝑓 ( 𝑥)= 𝑒 𝜎 √2 𝜋 for Basic properties Normal distributions: i.e. the total area under the normal curve above x-axis is 1. iii. Normal curve is symmetrical about its mean. iv. The mean, mode and median coincide for this distribution. 12/13/2024 Dr. Anil Agarwal Unit-III 45  Cont…(CO3) Standard form of the normal distribution: The probability density function for the normal distribution in standard form 1 is given − 1 by 𝑧 2 𝑓 ( 𝑧 )= 𝑒 2 √2𝜋 By taking , standard normal curve is formed. The total area under the curve is 1 and it is divided into two parts by z=0. 12/13/2024 Dr. Anil Agarwal Unit-III 46  Mean and Variance of Normal distribution(CO3) Mean of Normal Distribution: A.M. of continuous distribution f(x) is given by because So Let so that therefore 12/13/2024 Dr. Anil Agarwal Unit-III 47  Cont…(CO-3) Because so in above equation 12/13/2024 Dr. Anil Agarwal Unit-III 48  Cont…(CO-3) Variance of Normal distribution : Let so thattherefore 12/13/2024 Dr. Anil Agarwal Unit-III 49  Cont... (CO3) In equation(1) Variance s.d. of normal distribution is. 12/13/2024 Dr. Anil Agarwal Unit-III 50  Problem based on Normal distribution(CO3) Q1. A sample of 100 dry battery cells tested to find the length of the life produced the following results : Assuming the data to be normally distributed, what percentage of battery cells are expected to have life i. More than 15 hours ii. Less than 6 hours iii. b Solution: x denotes the life of dry battery cells. And i. When 12/13/2024 Dr. Anil Agarwal Unit-III 51  Cont…(CO3) Therefore ii. When Therefore 12/13/2024 Dr. Anil Agarwal Unit-III 52  Cont…(CO3) iii. When 12/13/2024 Dr. Anil Agarwal Unit-III 53  Daily Quiz(CO3) Q1. In a distribution exactly Normal, 31% of the items are under 45 and 8% are over 64. What are the mean and Standard deviation of this Distribution? It is given that if then 12/13/2024 Dr. Anil Agarwal Unit-III 54  Exponential Distribution(CO3) Exponential distribution: A continuous random variable which has the following Pdf , Where is a parameter. is called exponential distribution. Mean of Exponential distribution: We know that mean is given by Mean = , which is a Gamma function 12/13/2024 Dr. Anil Agarwal Unit-III 55  Exponential Distribution(CO3) Variance of Exponential distribution: We know that the variance is given by Variance ……….(1) Now , which is a Gamma function From (1), Variance Hence Variance 12/13/2024 Dr. Anil Agarwal Unit-III 56  Exponential Distribution(CO3) Q1. The length of Telephone conversation is an exponential variate with mean 3minutes. Find the Probability that call (a) End in less than 3 minutes. (b) tabes between 3 to 5 minutes. Solution. Given, Mean 12/13/2024 Dr. Anil Agarwal Unit-III 57  Daily Quiz(CO3) Q1. The length of Telephone conversation is an exponential variate with mean 7 minutes. Find the Probability that call (a) End in less than 7 minutes. (b) tabels between 7 to 10 minutes. 12/13/2024 Dr. Anil Agarwal Unit-III 58  Weekly Assignment(CO3) Q1. In a distribution exactly Normal, 31% of the items are under 45 and 8% are over 64. What are the mean and Standard deviation of this Distribution? It is given that if then Q2. The life of army shoes is normally distributed with mean 8 months and standard deviation 2 months. If 5000 airs are insured, how many pairs would you expected to need replacement after 12 months? Q3. Find the mean and variance of the Binomial, Poisson and Normal distribution. 12/13/2024 Dr. Anil Agarwal Unit-III 59  Weekly Assignment(CO3) Q4. The probability that a pen manufactured by a company will be defective is If 12 such pens are manufactured, find the probability that i. Exactly two will be defective ii. At least two will be defective iii. None will be defective. Q5. It is given that 2% of the electric bulbs manufactured by a company are defective. Using Poisson distribution find the probability that a sample of 200 bulbs will contain iv. No defective bulb v. Two defective bulbs vi. At most three defective bulbs. 12/13/2024 Dr. Anil Agarwal Unit-III 60  Topic Video Links, Youtube & NPTEL Video Video Link: Binomial distribution https://youtu.be/6pZXCcoeYiU Poisson Distribution https:// youtu.be/izT2QpldbnU Normal distribution https://youtu.be/UaLNsZQK8fo Suggested Video inks: Bay’s theorem https://www.youtube.com/watch?v=GSEu5hn2q98 Conditional probability https://youtu.be/hxn_QwwWZBQ 12/13/2024 Dr. Anil Agarwal Unit-III 61  MCQs(CO3) Q1. Suppose that a random variable has normal distribution with mean 9 and variance 9. Then the value of such that is-(Given that ) i. 12 ii. 1 iii. 10 iv. None of these Q2.The life of army shoes is normally distributed with mean 8 months and standard deviation 2 months. If 5000 pairs are issued, how many pairs would be expected to need replacement after 12 months are- (Given that ) v. 114 vi. 4886 vii.115 12/13/2024 Dr. Anil Agarwal Unit-III 62  MCQs(CO3) a. None of these Q3. Suppose that a book of 600 pages contains 40 printing mistakes. Assume that these errors are randomly distributed throughout the book and , the number of errors per page has a Poisson distribution then what is probability that 10 pages selected at random will be free of errors i. 0.54 ii. 0.15 iii. 0.51 iv. None of these Dr. Anil Agarwal Unit-III 12/13/2024 63  MCQs(CO3) Q4. The probability that a bomb dropped from a plane will strike the target is. If six bombs are dropped then the probability that exactly two will strike the target is i. 0.345 ii. 0.145 iii. 0.245 iv. None of these Q5. For the standard normal variate mean and variance are- v. 0,1 vi. 1,0 12/13/2024 Dr. Anil Agarwal Unit-III 64  Glossary(CO3) Q1. The distribution of the number of road accidents per day in a city is poisson with mean 4. find the number of days out of 100 days when there will be i. No accident ii. At least two accident iii. At most three accident iv. Between two and five accident Pick the correct option from glossary a. 91 b. 43 c. 39 d. 2 12/13/2024 Dr. Anil Agarwal Unit-III 65  Glossary(CO3) Q2.In 800 families with 4 children each ,how many families would be expected to have i. 2 boys and 2 girls ii. No girl iii. At least one boy iv. At most two girls (Assume equal probabilities for boys and girls) Pick the correct option from glossary a. 750 b. 550 c. 50 d. 300 12/13/2024 Dr. Anil Agarwal Unit-III 66  (Sessional + University) Question Paper Microsoft Word Microsoft Word Microsoft Word Document Document Document First Second Third Sessional Sessional Sessional Test Test Paper Test Paper Paper 12/13/2024 Dr. Anil Agarwal Unit-III 67  Expected Questions for External Exam Q.1 A bag contains 5Exam(CO3) white and 8 red balls. Two successive drawing of 3 balls are made such that (i) the balls are replaced before the second trial, and (ii)the balls are not replaced before the second trial. Find the probability that the first drawing will give 3 white and the second 3 red balls in each case. Q.2 The experience shows that 4 accidents occur in a plant on an average per month. Calculate the probabilities of less than 3 accidents in a certain month. Use Poisson distribution. (Given e-4=0.01832). Q.3 As a result of tests on 20,000 electric bulbs manufactured by a company it was found that the life time of a bulb was normally distributed with an average life of 2040 hours and standard deviation of 60 hours. On the basis of the information, 12/13/2024 Dr. Anil Agarwal estimate Unit-III the number 68 of  Expected Questions for External Exam Q4. In Exam(CO3) 800 families with 5 children each ,how many families would be expected to have i. 3 boys and 2 girls ii. No girl iii. At most 2 girls iv. Either 2 or 3 boys (Assume equal probabilities for boys and girls) Ans: 250, 25, 400, 25 Q5. What are the conditions necessary for a normal distribution to occur? With the help of a suitable diagram, list the chief properties of a normal distribution. 12/13/2024 Dr. Anil Agarwal Unit-III 69  Expected Questions for External Exam(CO3) Q6. In a test on 2000 electric bulbs,it was found that the life of a particular make, was normally distributed with an average life of 2040 hours and S.D. of 60 hours, estimate the number of bulbs likely to burn for i. More than 2150 hours ii. Less than 1950 hours iii. More than 1920 hours but less than 2160 hours Ans: 67,134,1909 Q7. In a distribution exactly Normal, 7% of the items are under 35 and 89% are under 63. What are the mean and Standard deviation of this Distribution? Ans: 50.3, 10.33 12/13/2024 Dr. Anil Agarwal Unit-III 70  Recap of Unit-III Students have taught the importance of the following topics…. With the concept of probability to evaluate probability distributions: Binomial Distribution Poisson Distribution Normal Distribution Exponential Distribution 12/13/2024 Dr. Anil Agarwal Unit-III 71  References Text Books Erwin Kreyszig, Advanced Engineering Mathematics, 9thEdition, John Wiley & Sons, 2006. P. G. Hoel, S. C. Port and C. J. Stone, Introduction to Probability Theory, Universal Book Stall, 2003(Reprint). S. Ross: A First Course in Probability, 6th Ed., Pearson Education India, 2002. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 3rd Ed., Wiley, 1968. 12/13/2024 Dr. Anil Agarwal Unit-III 72  References Reference Books B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 35th Edition, 2000. 2.T.Veerarajan : Engineering Mathematics (for semester III), Tata McGraw-Hill, New Delhi. R.K. Jain and S.R.K. Iyenger: Advance Engineering Mathematics; Narosa Publishing House, New Delhi. J.N. Kapur: Mathematical Statistics; S. Chand & Sons Company Limited, New Delhi. D.N.Elhance,V. Elhance & B.M. Aggarwal: Fundamentals of Statistics; Kitab Mahal Distributers, New Delhi. 12/13/2024 Dr. Anil Agarwal Unit-III 73  Thank You 12/13/2024 Dr. Anil Agarwal Unit-III 74