09 Binom Hyp Geo

# 09 Binom Hyp Geo - Random Variables Denoted X x element of...

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3/26/10 Random Variables Denoted X element of r.v. domain Random Variables are a function of the outcomes of an experiment Mapping of outcomes to 1 2 3 4 11 12 (1,1 ) (1,2 ) (3,1 ) (2,6 ) (2,1 ) (3,6 ) (6,6 ) x

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3/26/10 Probability Mass Functions Properties of Valid PMFs 1. Probabilities of all elements in domain should add to one 2. Probabilities must all be positive Thus fX(x) means P(X=x) for values x in the domain. This can be a little confusing. Here X is the random variable, while x is an algebraic
3/26/10 Cumulative Distribution Often times rather than wanting to know P(X=x) we will want to know P(X&x). This probability P(X&x) is known as the cumulative distribution function, or CDF. Given the PMF, one can obtain the CDF, and vice versa. = x z x X z f x F ) ( ) (

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3/26/10 Bernoulli Trials Models an experiment with 2 possible outcomes e.g. call them success and failure – {0,1} Each trial independent from previous Constant success rates Experiment Single roll of a fair, six-sided die
3/26/10 Bernoulli Distribution PX(x) =px (1-p)1-x for x"{0,1} Success: PX(x) = p What if we had more than 1 trial? What is the probability of one 6 in three rolls?

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3/26/10 The Binomial Distribution What is the probability of rolling exactly 2 sixes in ten rolls of a fair die. Let “6” denote the outcome is a six, and “N” denote that the outcome is not a six. Two example outcomes with exactly two sixes are NNN6NN6NNN
3/26/10 The Binomial Distribution I can generalize this even further. Suppose the probability of a “successful" outcome (e.g. a “6”) was not 1/6 but “p”. Also the probability of an "unsuccessful" outcome is 1-p. Then I can model any situation in which 10 independent Bernoulli trials occur.

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## This note was uploaded on 03/26/2010 for the course IE 111 taught by Professor Storer during the Spring '07 term at Lehigh University .

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09 Binom Hyp Geo - Random Variables Denoted X x element of...

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