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**Unformatted text preview: **Stat 400 Lecture 6 Spring 2012 Review (2.1,2.2,2.3) Mathematical expectation (mean,variance,standard de- viation, moment) Properties of mathematical expectation (mean,variance) Sample mean, variance Todays Lecture (2.4, 2.5) Bernoulli distribution and its mean, variance. Binomial distribution and its p.m.f., properties. Cumulative distribution function and its properties. Geometric distribution and Negative binomial distri- bution 1 Stat 400 Lecture 6 Spring 2012 Bernoulli distribution Definition: A random experiment is called a set of Bernoulli trials if it consists of several trials such that Each trial has only 2 possible outcomes (usually called Success and Failure). The probability of success p , remains constant for all trials; The trials are independent, i.e. the event success in trial i does not depend on the outcome of any other trials. Examples: Repeated tossing of a fair die: success=6, failure=not 6. Each toss is a Bernoulli trial with P ( success ) = Definition: The random variable X is called a Bernoul- li random variable if it takes only 2 values, 0 and 1. The p.m.f. f X ( x ) = p if x = 1 1- p if x = 0 E ( X ) V ar ( X ) 2 Stat 400 Lecture 6 Spring 2012 Binomial distribution Definition: Let X be the number of successes in n inde- pendent Bernoulli trials each with probability of success...

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