Probability And Statistics Finals Notes

Statistics notes Andre Sostar November 27, 2014 Contents 1 Discrete Random Variables 5 Bernoulli Random Variables 6 The Binomial Distribution 7 The Geometric Distribution 8 The Negative Binomial Distribution 8 The Hypergeometric Distribution 9 The Poisson Distribution 9 2 Continuous Random Variables 10 The Exponential Density 11 The Gamma Density 11 The Beta Density 12 The Normal Distribution 12 3 Functions of a Random Variable 13 The cdf Method 13 The Transformation Method 13 1 Generating pseudorandom variables 4 The Expected Value 13 15 The Expected Value of a random variable 15 Calculating Expected Values 15 Expected values of functions of random variables 15 5 The Variance 16 The Variance of a random variable 16 Chebyshev's inequality 17 6 Moment Generating Functions 18 Moments 18 Moment Generating Functions 18 7 Joint and Marginal Distributions 20 Multivariate Random Variables 20 Continuous r.v. joint pdf 20 Marginal cdf and marginal pdf 21 8 Independent Random Variables 23 Independent Random Variables 23 Conditional probability mass function 23 Conditional probability density function 24 The Law of Total Probability 24 9 Functions of Jointly Distributed r.v's 25 The Method of Distribution Functions 25 The Distribution of the Sum 25 2 The Method of Moment Generating Functions 25 The Moment Generating Functions for sums and means of i.i.d. r.v. 25 Expected values functions of jointly distributed r.v. 10 Covariance and Correlation 26 27 The concept of dependence 27 Linear dependency 27 Covariance 27 Covariance 27 The Correlation Coecient 28 Covariance of linear combinations 28 11 Conditional expectations 30 Conditional expectations 30 Conditional variance 30 12 The Law of Large Numbers 31 Unbiasedness 31 Consistency 31 The Law of Large Numbers 31 13 Convergence in distribution 32 Convergence in distribution 32 Theorem proving Convergence in distribution 32 Standardizing 32 14 The Central Limit Theorem 33 3 The Central Limit Theorem 15 Other Important Laws 33 34 The Multiplicative Law 34 The Multiplicative Law (bivariate) 34 The conditional pmf/pdf 34 Hierarchic Models 34 Independent Addition Law 34 4 1 Discrete Random Variables Denition: The probability that Y takes on the value y, P (Y = y), is dened as the sum of the probabilities of all sample points in S that are assigned the value y. We will sometimes denote P (Y = y) by p(y). For any discrete probability distribution, the following must be true: 1. 0 [?] p(y) [?] 1 for all y. 2. [?] p(y) = 1, where the summation is over all values of y with nonzero probability. Denition: A random variable on is a real valued function X : - R, that is a function that assigns a real value to each possible outcome in . Denition: A discrete random variable is a random variable that can take on only a nite or at most a countably innite number of values. Denition: The probability mass function (pmf ), is the probability measure on the sample space that determines the probabilities of the various values of X; is those values are denoted by x1 , x2 , * * * , then there is a function p such that p(xi ) = P (X = xi ) and [?] p(xi ) = 1. i How to determine if the discrete random variable is one of the common types. Question 1. Is it possible to interpret the random experiment as sampling from an urn with only two types of balls (e.g. successes and failures)? Question 2. If so, is the draw done with or without replacement, i.e. is the probability of success always the same? Question 3. Is the number of balls to be drawn xed, or are we to draw balls until some criterion is met? Question 4. In what way does our random variable evaluate the outcome of the random experiment, i.e. the drawn balls? 5 The cumulative distribution function (cdf ) of a random variable, is dened to be F (x) = P (X [?] x), -[?]Buy the full version of these notes or essay plans and more in our Probability and Statistics Notes.