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Probability and Counting Why study probability? Sample spaces and Pebble World Naive definition of probability How to count Story proofs Non-naive definition of probability Recap R Exercises Conditional Probability The importance of thinking conditionally Definition and intuition Bayes'' rule and the law of total probability Conditional probabilities are probabilities Independence of events Coherency of Bayes'' rule Conditioning as a problem-solving tool Pitfalls and paradoxes Recap R Exercises Random Variables and Their Distributions Random variables Distributions and probability mass functions Bernoulli and Binomial Hypergeometric Discrete Uniform Cumulative distribution functions Functions of random variables Independence of rvs Connections between Binomial and Hypergeometric Recap R Exercises Expectation Definition of expectation Linearity of expectation Geometric and Negative Binomial Indicator rvs and the fundamental bridge Law of the unconscious statistician (LOTUS) Variance Poisson Connections between Poisson and Binomial *Using probability and expectation to prove existence Recap R Exercises Continuous Random Variables Probability density functions Uniform Universality of the Uniform Normal Exponential Poisson processes Symmetry of iid continuous rvs Recap R Exercises Moments Summaries of a distribution Interpreting moments Sample moments Moment generating functions Generating moments with MGFs Sums of independent rvs via MGFs *Probability generating functions Recap R Exercises Joint Distributions Joint, marginal, and conditional D LOTUS Covariance and correlation Multinomial Multivariate Normal Recap R Exercises Transformations Change of variables Convolutions Beta Gamma Beta-Gamma connections Order statistics Recap R Exercises Conditional Expectation Conditional expectation given an event Conditional expectation given an rv Properties of conditional expectation *Geometric interpretation of conditional expectation Conditional variance Adam and Eve examples Recap R Exercises Inequalities and Limit Theorems Inequalities Law of large numbers Central limit theorem Chi-Square and Student-t Recap R Exercises Markov Chains Markov property and transition matrix Classification of states Stationary distribution Reversibility Recap R Exercises Markov Chain Monte Carlo Metropolis-Hastings Recap R Exercises Poisson Processes Poisson processes in one dimension Conditioning, superposition, thinning Poisson processes in multiple dimensions Recap R Exercises A Math A Sets A Functions A Matrices A Difference equations A Differential equations A Partial derivatives A Multiple integrals A Sums A Pattern recognition A Common sense and checking answers B R B Vectors B Matrices B Math B Sampling and simulation B Plotting B Programming B Summary statistics B Distributions C Table of distributions Bibliography Index >R Exercises Random Variables and Their Distributions Random variables Distributions and probability mass functions Bernoulli and Binomial Hypergeometric Discrete Uniform Cumulative distribution functions Functions of random variables Independence of rvs Connections between Binomial and Hypergeometric Recap R Exercises Expectation Definition of expectation Linearity of expectation Geometric and Negative Binomial Indicator rvs and the fundamental bridge Law of the unconscious statistician (LOTUS) Variance Poisson Connections between Poisson and Binomial *Using probability and expectation to prove existence Recap R Exercises Continuous Random Variables Probability density functions Uniform Universality of the Uniform Normal Exponential Poisson processes Symmetry of iid continuous rvs Recap R Exercises Moments Summaries of a distribution Interpreting moments Sample moments Moment generating functions Generating moments with MGFs Sums of independent rvs via MGFs *Probability generating functions Recap R Exercises Joint Distributions Joint, marginal, and conditional D LOTUS Covariance and correlation Multinomial Multivariate Normal Recap R Exercises Transformations Change of variables Convolutions Beta Gamma Beta-Gamma connections Order statistics Recap R Exercises Conditional Expectation Conditional expectation given an event Conditional expectation given an rv Properties of conditional expectation *Geometric interpretation of conditional expectation Conditional variance Adam and Eve examples Recap R Exercises Inequalities and Limit Theorems Inequalities Law of large numbers Central limit theorem Chi-Square and Student-t Recap R Exercises Markov Chains Markov property and transition matrix Classification of states Stationary distribution Reversibility Recap R Exercises Markov Chain Monte Carlo Metropolis-Hastings Recap R Exercises Poisson Processes Poisson processes in one dimension Conditioning, superposition, thinning Poisson processes in multiple dimensions Recap R Exercises A Math A Sets A Functions A Matrices A Difference equations A Differential equations A Partial derivatives A Multiple integrals A Sums A Pattern recognition A Common sense and checking answers B R B Vectors B Matrices B Math B Sampling and simulation B Plotting B Programming B Summary statistics B Distributions C Table of distributions Bibliography Index ;P>Law of the unconscious statistician (LOTUS) Variance Poisson Connections between Poisson and Binomial *Using probability and expectation to prove existence Recap R Exercises Continuous Random Variables Probability density functions Uniform Universality of the Uniform Normal Exponential Poisson processes Symmetry of iid continuous rvs Recap R Exercises Moments Summaries of a distribution Interpreting moments Sample moments Moment generating functions Generating moments with MGFs Sums of independent rvs via MGFs *Probability generating functions Recap R Exercises Joint Distributions Joint, marginal, and conditional D LOTUS Covariance and correlation Multinomial Multivariate Normal Recap R Exercises Transformations Change of variables Convolutions Beta Gamma Beta-Gamma connections Order statistics Recap R Exercises Conditional Expectation Conditional expectation given an event Conditional expectation given an rv Properties of conditional expectation *Geometric interpretation of conditional expectation Conditional variance Adam and Eve examples Recap R Exercises Inequalities and Limit Theorems Inequalities Law of large numbers Central limit theorem Chi-Square and Student-t Recap R Exercises Markov Chains Markov property and transition matrix Classification of states Stationary distribution Reversibility Recap R Exercises Markov Chain Monte Carlo Metropolis-Hastings Recap R Exercises Poisson Processes Poisson processes in one dimension Conditioning, superposition, thinning Poisson processes in multiple dimensions Recap R