Probability and Statistics for Physical Sciences,

1. Statistics, Experiments, and Data
1.1 Experiments and Observations
1.2 Random Variables and Sampling
1.3 Displaying Data
1.4 Summarizing Data Numerically
1.4.1 Measures of location
1.4.2 Measures of spread
1.4.3 More than one variable: correlation
1.5 Large Samples
1.6 Experimental Errors

2. Probability
2.1 Sample Spaces and Events
2.2 Axioms and Calculus of Probability
2.3 Conditional and Marginal Probabilities
2.4 Permutations and Combinations
2.5 The Meaning of Probability
2.5.1 Frequency interpretation
2.5.2 Subjective interpretation

3. Probability Distributions I : Basic Concepts
3.1 Random Variables
3.2 Single Variable
3.2.1 Probability distributions
3.2.2 Expectation values
3.2.3 Moment-generating, and characteristic functions
3.3 Several Variables
3.3.1 Joint probability distributions
3.3.2 Marginal and conditional distributions
3.3.3 Expectation values of sums and products
3.3.4 Moments and expectation values
3.4 Functions of a Random Variable

4. Probability Distributions II : Examples
4.1 Continuous Variables
4.1.1 Uniform distribution
4.1.2 Univariate normal (Gaussian) distribution
4.1.3 Multivariate normal distribution
4.1.4 Bivariate normal distribution
4.1.5 Exponential and gamma distributions
4.1.6 Cauchy distribution
4.2 Discrete Variables
4.2.1 Binomial distribution
4.2.2 Multinomial distribution
4.2.3 Poisson distribution

5. Sampling and Estimation
5.1 Random Samples and Estimators
5.1.1 Samplng distributions
5.1.2 Properties of point estimators
5.2 Estimators for the Mean, Variance, and Covariance
5.3 Laws of Large Numbers and the Central Limit Theorem
5.4 Experimental Errors
5.4.1 Propagation of errors
5.5 Monte Carlo Method and Simulations
5.5.1 Monte Carlo method for integration
5.5.2 Simulations

6. Sampling Distributions Associated with the Normal Distribution
6.1 Chi-Squared Distribution
6.2 Student’s t Distribution
6.3 F Distribution
6.4 Relations Between X2, t and F Distributions

7. Parameter Estimation I: Maximum Likelihood and Minimum Variance
7.1 Estimation of a Single Parameter
7.2 Variance of an Estimator
7.2.1 Approximate methods
7.3 Simultaneous Estimation of Several Parameters
7.4 Minimum Variance
7.4.1 Parameter estimation
7.4.2 Minimum variance bound

8. Parameter Estimation II : Least-Squares and Other Methods
8.1 Unconstrained Linear Least-Squares
8.1.1 General solution for the parameters
8.1.2 Errors on the parameter estimates
8.1.3 Quality of the fit
8.1.4 Orthogonal polynomials
8.1.5 Fitting a straight line
8.1.6 Combining experimental data
8.2 Linear Least-Squares with Constraints
8.3 Non-Linear Least-Squares
8.4 Other Methods
8.4.1 Minimum chi-squared
8.4.2 Method of moments
8.4.3 Bayes’ estimators

9. Interval Estimation
9.1 Confidence Intervals : Basic Ideas
9.2 Confidence Intervals : General Method
9.3 Normal Distribution
9.3.1 Confidence intervals for the mean
9.3.2 Confidence intervals for the variance
9.3.3 Confidence intervals for the ratio of variances
9.3.4 Confidence regions for the mean and variance
9.4 Poisson Distribution
9.5 Large Samples
9.6 Confidence Intervals Near Boundaries
9.7 Bayesian Confidence Intervals

10. Hypothesis Testing I : Parameters
10.1 Statistical Hypotheses
10.2 General Hypotheses : Likelihood Ratios
10.2.1 Simple hypothesis : one simple alternative
10.2.2 Composite hypotheses
10.3 Normal Distribution
10.3.1 Basic ideas
10.3.2 Specific tests
10.4 Other Distributions
10.5 Analysis of Variance
10.6 Bayesian Hypothesis Testing

11. Hypothesis Testing II : Other Tests
11.1 Goodness-Of-Fit Tests
11.1.1 Discrete distributions
11.1.2 Continuous distributions
11.1.3 Linear hypotheses
11.2 Tests for Independence
11.3 Nonparametric Tests
11.3.1 Sign test
11.3.2 Signed-rank test
11.3.3 Rank-sum test
11.3.4 Run test
11.3.5 Rank correlation coefficient

Appendix A. Miscellaneous Mathematics
A.1 Matrix Algebra
A.2 Classical Theory of Minima
A.3 Delta function
A.4 Distribution of the Mean of a Poisson Sample
A.5 The Chebyshev Inequality

Appendix B. Optimization of Nonlinear Functions
B.1 General Principles
B.2 Unconstrained Minimization of Functions of One variable
B.3 Unconstrained Minimization of Multivariable Functions
B.3.1 Direct search methods
B.3.2 Gradient methods
B.4 Constrained Optimization

Appendix C. Statistical Tables
C.1 Normal Distribution
C.2 Binomial Distribution
C.3 Poisson Distribution
C.4 Chi-squared Distribution
C.5 Student’s t Distribution
C.6 F distribution
C.7 Signed-Rank Test
C.8 Rank-Sum Test
C.9 Run Test
C.10 Rank Correlation Coefficient

Appendix D. Answers to Selected Problems