Lecture05 - Binomial Setting Probability Theory The...

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Probability Theory The Binomial and Poisson Distributions Sections 5.2 and 5.3 © 2009 W. H. Freeman and Company Binomial Setting Binomial distributions are models for some categorical variables, typically representing the number of successes in a series of n trials. The observations must meet these requirements: ! The total number of observations n is fixed in advance. ! The outcomes of all n observations are statistically independent. ! Each observation falls into just one of 2 categories: success and failure. ! All n observations have the same probability of “success,” p . We record the next 50 births at a local hospital. Each newborn is either a boy or a girl; each baby is either born on a Sunday or not. The distribution of the count X of successes in the binomial setting is the binomial distribution with parameters n and p: B ( n,p ). ! The parameter n is the total number of observations. ! The parameter p is the probability of success on each observation. ! The count of successes X can be any whole number between 0 and n . A coin is flipped 10 times. Each outcome is either a head or a tail. The variable X is the number of heads among those 10 flips, our count of “successes.” On each flip, the probability of success, “head,” is 0.5. The number X of heads among 10 flips has the binomial distribution B ( n = 10, p = 0.5). Binomial Distribution Applications for binomial distributions Binomial distributions describe the possible number of times that a particular event will occur in a sequence of observations. They are used when we want to know about the occurrence of an event, not its magnitude. ! In a clinical trial, a patient’s condition may improve or not. We study the number of patients who improved, not how much better they feel. ! Is a person ambitious or not? The binomial distribution describes the number of ambitious persons, not how ambitious they are. ! In quality control we assess the number of defective items in a lot of goods, irrespective of the type of defect.
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Binomial Probabilities The number of ways of arranging k successes in a series of n observations (with constant probability p of success) is the number of possible combinations (unordered sequences). This can be calculated with the binomial coefficient : Where k = 0, 1, 2, . .., or n. Binomial formulas " The binomial coefficient “ n _choose_ k ” uses the factorial notation ! ”. " The factorial n ! for any strictly positive whole number n is: n ! = n ! ( n " 1) ! ( n " 2) ! # # # ! 3 ! 2 ! 1 ! For example: 5 ! = 5 ! 4 ! 3 ! 2 ! 1 = 120 ! Note that 0! = 1. Calculations for binomial probabilities The binomial coefficient counts the number of ways in which k successes can be arranged among n observations. The
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This note was uploaded on 02/16/2010 for the course STAT 212 taught by Professor Holt during the Fall '08 term at UVA.

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Lecture05 - Binomial Setting Probability Theory The...

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