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# Note for Probability and Statistics - PS By JNTU Heroes

• Probability and Statistics - PS
• Note
• Jawaharlal Nehru Technological University Anantapur (JNTU) College of Engineering (CEP), Pulivendula, Pulivendula, Andhra Pradesh, India - JNTUACEP
• 15 Topics
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PROBABILITY AND STATISTICS lecture notes

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Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Certain event. Impossible event. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Contrary events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Compatible events. Incompatible events. . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Event implied by another event. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Operations with events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7 Sample space of an experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.8 Frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.9 Equally possible events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.10 Probability of an event. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.11 Finite sample space. Elementary event. . . . . . . . . . . . . . . . . . . . . . . . 12 1.12 Axiomatic definition of probability . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.13 Independent and dependent events. . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.14 Conditional probability 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 One-dimensional discrete random variables . . . . . . . . . . . . . . . . . . . . . 23 1.16 The distribution function of a discrete one-dimensional random variable . . . . . 26 1.17 Two-dimensional discrete random variables (random vectors) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.18 The distribution function of a random vector . . . . . . . . . . . . . . . . . . . . 31 1.19 Expected value. Variance. Moments. (for discrete one-dimensional random variables) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.20 Covariance. Correlation coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.21 Convergence of sequences of random variables. . . . . . . . . . . . . . . . . . . . 38 1.22 Law of large numbers 39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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4 CONTENTS 1.23 Binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.24 The Poisson distribution as an approximation of the binomial distribution . . . . 43 1.25 The multinomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.26 Geometric distribution. Negative binomial distribution . . . . . . . . . . . . . . . 47 1.27 Continuous random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.28 The distribution function for the continuous random variables. Probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.29 The expected values and the variance of a continuous random variable . . . . . . 51 1.30 The normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2 STATISTICS 55 2.1 What is Statistics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.4 Determining the frequency and grouping the data . . . . . . . . . . . . . . . . . . 60 2.5 Data presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.6 Parameters and statistics of the central tendency . . . . . . . . . . . . . . . . . . 67 2.7 Parameters and statistics of dispersion . . . . . . . . . . . . . . . . . . . . . . . . 70 2.8 Factorial parameters and statistics of the variance . . . . . . . . . . . . . . . . . 72 2.9 Parameters and statistics of position . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.10 The sampling distribution of the sample statistics . . . . . . . . . . . . . . . . . . 74 2.11 The central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.12 An application of the central limit theorem . . . . . . . . . . . . . . . . . . . . . 79 2.13 Point estimation for a parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.14 Generalities regarding the problem of hypothesis testing . . . . . . . . . . . . . . 81 2.15 Hypothesis test: A classical approach . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.16 Hypothesis test: a probability-value approach . . . . . . . . . . . . . . . . . . . . 89 2.17 Statistical inference about the population mean when the standard deviation is not known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.18 Inferences about the variance and the estimation of the variance . . . . . . . . . 98 2.19 Generalities about correlation. Linear correlation . . . . . . . . . . . . . . . . . . 104 2.20 Linear correlation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.21 Inferences about the linear correlation coefficient . . . . . . . . . . . . . . . . . . 113