Probability and Statistics: A Course for Physicists and Engineers

Preface
Acknowledgement
List of Tables
List of Symbols

1 Random phenomena
1.1 Introduction
1.1.1 Waiting time
1.1.2 Random events
1.2 A random experiment
1.3 Venn diagrams
1.4 Probability or chance of occurrence of a random event
1.4.1 Postulates for probability
1.5 How to assign probabilities to individual events?
1.5.1 Buffon’s “clean tile problem”

2 Probability
2.1 Introduction
2.2 Permutations
2.3 Combinations
2.4 Sum Σ and product Π notation
2.4.1 The product notation or pi notation
2.5 Conditional probabilities
2.5.1 The total probability law
2.5.2 Bayes’ rule
2.5.3 Entropy

3 Random variables
3.1 Introduction
3.2 Axioms for probability/density function and distribution functions
3.2.1 Axioms for a distribution function
3.2.2 Mixed cases
3.3 Some commonly used series
3.3.1 Exponential series
3.3.2 Logarithmic series
3.3.3 Binomial series
3.3.4 Trigonometric series
3.3.5 A note on logarithms

4 Expected values
4.1 Introduction
4.2 Expected values
4.2.1 Measures of central tendency
4.2.2 Measures of scatter or dispersion
4.2.3 Some properties of variance
4.3 Higher moments
4.3.1 Moment generating function
4.3.2 Moments and Mellin transforms
4.3.3 Uniqueness and the moment problem

5 Commonly used discrete distributions
5.1 Introduction
5.2 Bernoulli probability law
5.3 Binomial probability law
5.4 Geometric probability law
5.5 Negative binomial probability law
5.6 Poisson probability law
5.6.1 Poisson probability law from a process
5.7 Discrete hypergeometric probability law
5.8 Other commonly used discrete distributions

6 Commonly used density functions
6.1 Introduction
6.2 Rectangular or uniform density
6.3 A two-parameter gamma density
6.3.1 Exponential density
6.3.2 Chi-square density
6.4 Generalized gamma density
6.4.1 Weibull density
6.5 Beta density
6.6 Laplace density
6.7 Gaussian density or normal density
6.7.1 Moment generating function of the normal density
6.8 Transformation of variables
6.9 A note on skewness and kurtosis
6.1 0Mathai’s pathway model
6.1 0.1 Logistic model
6.1 1Some more commonly used density functions

7 Joint distributions
7.1 Introduction
7.2 Marginal and conditional probability/density functions
7.2.1 Geometrical interpretations of marginal and conditional distributions
7.3 Statistical independence of random variables
7.4 Expected value
7.4.1 Some properties of expected values
7.4.2 Joint moment generating function
7.4.3 Linear functions of random variables
7.4.4 Some basic properties of the correlation coefficient
7.5 Conditional expectations
7.5.1 Conditional expectation and prediction problem
7.5.2 Regression
7.6 Bayesian procedure
7.7 Transformation of variables

8 Some multivariate distributions
8.1 Introduction
8.2 Some multivariate discrete distributions
8.2.1 Multinomial probability law
8.2.2 The multivariate hypergeometric probability law
8.3 Some multivariate densities
8.3.1 Type-1 Dirichlet density
8.3.2 Type-2 Dirichlet density
8.4 Multivariate normal or Gaussian density
8.4.1 Matrix-variate normal or Gaussian density
8.5 Matrix-variate gamma density

9 Collection of random variables
9.1 Introduction
9.2 Laws of large numbers
9.3 Central limit theorems
9.4 Collection of dependent variables

10 Sampling distributions
10.1 Introduction
10.2 Sampling distributions
10.3 Sampling distributions when the population is normal
10.4 Student-t and F distributions
10.5 Linear forms and quadratic forms
10.6 Order statistics
10.6.1 Density of the smallest order statistic xn∶1
10.6.2 Density of the largest order statistic xn∶n
10.6.3 The density of the r-th order statistic xn∶r
10.6.4 Joint density of the r-th and s-th order statistics xn∶r and xn∶s

11 Estimation
11.1 Introduction
11.2 Parametric estimation
11.3 Methods of estimation
11.3.1 The method of moments
11.3.2 The method of maximum likelihood
11.3.3 Method of minimum Pearson’s X2 statistic or minimum chi-square method
11.3.4 Minimum dispersion method
11.3.5 Some general properties of point estimators
11.4 Point estimation in the conditional space
11.4.1 Bayes’ estimates
11.4.2 Estimation in the conditional space: model building
11.4.3 Some properties of estimators
11.4.4 Some large sample properties of maximum likelihood estimators
11.5 Density estimation
11.5.1 Unique determination of the density/probability function
11.5.2 Estimation of densities

12 Interval estimation
12.1 Introduction
12.2 Interval estimation problems
12.3 Confidence interval for parameters in an exponential population
12.4 Confidence interval for the parameters in a uniform density
12.5 Confidence intervals in discrete distributions
12.5.1 Confidence interval for the Bernoulli parameter p
12.6 Confidence intervals for parameters in N(μ, σ2)
12.6.1 Confidence intervals for μ
12.6.2 Confidence intervals for σ2 in N(μ, σ2)
12.7 Confidence intervals for linear functions of mean values
12.7.1 Confidence intervals for mean values when the variables are dependent
12.7.2 Confidence intervals for linear functions of mean values when there is statistical independence
12.7.3 Confidence intervals for the ratio of variances

13 Tests of statistical hypotheses
13.1 Introduction
13.2 Testing a parametric statistical hypothesis
13.2.1 The likelihood ratio criterion or the λ-criterion
13.3 Testing hypotheses on the parameters of a normal population N(μ, σ2)
13.3.1 Testing hypotheses on μ in N(μ, σ2) when σ2 is known
13.3.2 Tests of hypotheses on μ in N(μ, σ2) when σ2 is unknown
13.3.3 Testing hypotheses on σ2 in a N(μ, σ2)
13.4 Testing hypotheses in bivariate normal population
13.5 Testing hypotheses on the parameters of independent normal populations
13.6 Approximations when the populations are normal
13.6.1 Student-t approximation to normal
13.6.2 Approximations based on the central limit theorem
13.7 Testing hypotheses in binomial, Poisson and exponential populations
13.7.1 Hypotheses on p, the Bernoulli parameter
13.7.2 Hypotheses on a Poisson parameter
13.7.3 Hypotheses in an exponential population
13.8 Some hypotheses on multivariate normal
13.8.1 Testing the hypothesis of independence
13.9 Some non-parametric tests
13.9.1 Lack-of-fit or goodness-of-fit tests
13.9.2 Test for no association in a contingency table
13.9.3 Kolmogorov–Smirnov statistic Dn
13.9.4 The sign test
13.9.5 The rank test
13.9.6 The run test

14 Model building and regression
14.1 Introduction
14.2 Non-deterministic models
14.2.1 Random walk model
14.2.2 Branching process model
14.2.3 Birth and death process model
14.2.4 Time series models
14.3 Regression type models
14.3.1 Minimization of distance measures
14.3.2 Minimum mean square prediction
14.3.3 Regression on several variables
14.4 Linear regression
14.4.1 Correlation between x1 and its best linear predictor
14.5 Multiple correlation coefficient ρ1.(2…k)
14.5.1 Some properties of the multiple correlation coefficient
14.6 Regression analysis versus correlation analysis
14.6.1 Multiple correlation ratio
14.6.2 Multiple correlation as a function of the number of regressed variables
14.7 Estimation of the regression function
14.7.1 Estimation of linear regression of y on x
14.7.2 Inference on the parameters of a simple linear model
14.7.3 Linear regression of y on x1, … , xk
14.7.4 General linear model

15 Design of experiments and analysis of variance
15.1 Introduction
15.2 Fully randomized experiments
15.2.1 One-way classification model as a general linear model
15.2.2 Analysis of variance table or ANOVA table
15.2.3 Analysis of individual differences
15.3 Randomized block design and two-way classifications
15.3.1 Two-way classification model without interaction
15.3.2 Two-way classification model with interaction
15.4 Latin square designs
15.5 Some other designs
15.5.1 Factorial designs
15.5.2 Incomplete block designs
15.5.3 Response surface analysis
15.5.4 Random effect models

16 Questions and answers
16.1 Questions and answers on probability and random variables
16.2 Questions and answers on model building
16.3 Questions and answers on tests of hypotheses

Tables of percentage points
References
Index