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Probability and Statistics Basics

Probability and Statistics BasicsKevin Kircher Cornell MAE Spring 14 These notes summarize some basic Probability and Statistics material. The primarysources areA Modern Introduction to Probability and Statisticsby Dekking, Kraaikamp,Lopuha a and Meester,Introduction to Probabilityby Dimitri Bertsekas, and the lectures ofProfs. Gennady Samorodnitsky and Mark Probability31 Outcomes, Events and Probability32 conditional Probability and Independence53 Discrete Random Variables74 Continuous Random Variables105 The Normal Distribution136 Expectation and Variance177 Joint Distributions and Independence198 Covariance and Correlation229 Random Vectors2410 Transformations of Random Variables2611 The Law of Large Numbers2912 Moment Generating Functions3113 conditional Distributions32114 Order Statistics3515 The Central Limit Theorem3716 stochastic Processes39II Statistics4217 Numerical Data Summaries4218 Basic Statistical Models4319 Unbiased Estimators4420 Precision of an Estimator4521 Maximum Likelihood Estimation4722 Method of Moments Estimation4823 Bayesian Estimation4924 Least Squares Estimation5125 Minimum Mean Squared Error Estimation5226 Hypothesis Testing532 Part IProbability1 Outcomes.

16 Stochastic Processes39 II Statistics42 17 Numerical Data Summaries42 ... 2 Conditional Probability and Independence De nitions The conditional probability of A given C (C is called the conditioning event), provided P(C) >0, is ... Common Discrete Distributions Xhas the Bernoulli distribution Ber(p) with parameter 0 p 1 if its pmf is given by ...

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