A Modern Introduction to Probability and Statistics

Graham Upton is a retired Professor of Applied Statistics, formerly of the Department of Mathematical sciences at The University of Essex. He has published numerous books, including The Oxford Dictionary of Statistics, Data Analysis: A Gentle Introduction for Future Data Scientists, and Understanding Statistics with OUP.

• 1 Probability 1: Probability 1.1 Relative Frequency 1.2 Preliminary definitions 1.3 The probability scale 1.4 Probability with equally likely outcomes 1.5 The complementary event E' 1.6 Venn diagrams 1.7 Unions and intersections of events 1.8 Mutually exclusive events 1.9 Exhaustive events 1.10 Probability trees 1.11 Sample proportions and probability 1.12 Unequally likely possibilities 1.13 Physical independence 1.14 Orderings 1.15 Permutations and combinations 1.16 Sampling without replacement 1.17 Sampling without replacement 2: Conditional Probability 2.1 Notation 2.2 Statistical independence 2.3 Mutual and pairwise independence 2.4 The total probability theorem (The partition theorem) 2.5 Bayes> ' theorem 2.6 *The Monty Hall problem 3: Probability distributions 3.1 Notation 3.2 Probability distributions 3.3 The discrete uniform distribution 3.4 The Bernoulli distribution 3.5 The Binomial Distribution 3.6 Notation 3.7 <'Successes> ' and <'Failures> ' 3.8 The shape of the binomial distribution 3.9 The geometric distribution 3.10 The Poisson distribution and the Poisson process 3.11 The form of the distribution 3.12 Sums of Poisson random variables 3.13 The Poisson approximation to the binomial 3.14 The negative binomial distribution 3.15 The hypergeometric distribution 4: Expectations 4.1 Expectations of functions 4.2 The population variance 4.3 Sums of random variables 4.4 Mean and variance of common distributions 4.5 The expectation and variance of the sample mean 5: Continuous random variables 5.1 The probability density function (pdf) 5.2 The cumulative distribution function, F 5.3 Expectations for continuous variables 5.4 Obtaining f from F 5.5 The uniform (rectangular) distribution 5.6 The exponential distribution 5.7 *The beta distribution 5.8 *The gamma distribution 5.9 *Transformation of a random variable 6: The Normal Distribution 6.1 The general normal distribution 6.2 The use of tables 6.3 Linear combinations of independent normal random variables 6.4 The Central Limit Theorem 6.5 The normal distribution used as an approximation 6.6 *Proof that the area under the normal curve is 1 7: Distributions related to the normal distribution 7.1 The t distribution 7.2 The chi-squared distribution 7.3 The F distribution 8: *Generating functions 8.1 The probability generating function, G 8.2 The moment generating function 9: *Inequalities and laws 9.1 Markov> 's inequality 9.2 Chebyshev> 's inequality 9.3 The weak law of large numbers 9.4 The strong law of large numbers 10: Joint Distributions 10.1 Joint probability mass function 10.2 Marginal distributions 10.3 Conditional distributions 10.4 2 Statistics 11: Data sources 11.1 Data collection by observation 11.2 National Consuses 11.3 Sampling 11.4 Questionnaires 11.5 Questionnaire Design 12: Summarising data 12.1 A single variable 12.2 Two variables 12.3 More than two variables 12.4 Choosing which display to use 12.5 Dirty Data 13: General Summary Statistics 13.1 Measure of location: The mode 13.2 Measure of location: The mean 13.3 Measure of location: The mean of a frequency distribution 13.4 Measure of location: The mean of grouped data 13.5 Simplifying calculations 13.6 Measure of location: The median 13.7 Quantiles 13.8 Measures of spread: The Range and Inter-quartile Range 13.9 Boxplot 13.10 Deviations from the mean 13.11 The mean deviation 13.12 Measure of spread: The variance 13.13 Calculating the variance by hand 13.14 Measure of spread: The standard deviation 13.15 Variance and standard deviation for frequency distributions 13.16 Symmetric and skewed data 13.17 Standardising to a prescribed mean and standard deviation 13.8 *Calculating the combined mean and variance of several samples 13.19 Combining proportions 14: Point and interval estimation 14.1 Point estimates 14.2 Estimation methods 14.3 Confidence intervals 14.4 Confidence intervals with discrete distributions 14.5 One-sided confidence intervals 14.6 Confidence intervals for a variance 15: Single-sampled hypothesis tests 15.1 The null and alternative hypothesis 15.2 Critical regions and significance levels 15.3 The test procedure 15.4 Identifying two hypotheses 15.5 Tail probabilities: the p value approach 15.6 Hypothesis tests and confidence intervals 15.7 Hypothesis tests for a mean 15.8 Testing for normality 15.9 Hypothesis test for the variance of a normal distribution 5.10 Hypothesis tests with discrete distributions 5.11 Type I and Type II errors 5.12 Hypothesis tests for a proportion based on a small sample 15.13 Hypothesis tests for a Poisson mean based on a small sample 16: Two samples & paired samples 16.1 The comparison of two means 16.2 Confidence interval for the difference between two normal means 16.3 Paired samples 16.4 The comparison of the variances of two normal distributions 16.5 Confidence interval for a variance ratio 17: Goodness of fit 17.1 The chi-squared test 17.2 Small expected frequencies 17.3 Goodness of fit to prescribed distribution type 17.4 Comparing distribution functions 17.5 The dispersion test 17.6 Contingency tables 17.7 The 2 x 2 table: the comparison of two proportions 17.8 *Multi-way contingency tables 18: Correlation 18.1 The product-moment correlation coefficent 18.2 Nonsense correlation: storks and goosebury bushes 18.3 The ecological fallacy: Immigration and illiteracy 18.4 Simpson's paradox: Amputation or injection? 18.5 Rank correlation 19: Regression 19.1 The equation of a straight line 19.2 Why 'regression'? 19.3 The method of least squares 19.4 Transformations, extrapolation and outliers 19.5 Properties of the estimators 19.6 Analysis of Variance (ANOVA) 19.7 Multiple Regression 20 *The Bayesian approach