L9_rev2

# L9_rev2 - Binomial Distribution Assumptions n independent...

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Binomial Distribution Assumptions n independent trials trial i has dichotomous out- come (0 or 1, no or yes, fail- ure or success, etc.) denoted by X i . Called Bernoulli trials. probability of success on any trial is the same p. 1

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Let Y = # of successes. Y is called a binomial distribu- tion. P ( Y = y ) = g ( y ) = n y ! p y (1 - p ) n - y for y = 0 , 1 ,...,n. 2
Example. Multiple Choice Test. n = 10 questions. 4 possible answers. Ran- domly guess the correct answer. Y = #correct guesses Probabilities. n = 10 , p = 0 . 25 y P(Y=y) 0 0.0563 1 0.1877 2 0.2816 3 0.2503 4 0.1460 5 0.0584 6 0.0162 7 0.0031 8 0.0004 9 0.0000 10 0.0000 3

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0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 9 10 Find probability of at most 2 correct. P ( Y 2) = 0 . 0563 + 0 . 1877 + 0 . 2816 = 0 . 5256 Mean μ = np Variance σ 2 = np (1 - p ) 4
Brief peek at estimation Deﬁne b p = Y n the sample proportion. It is an estima- tor of p . Why? Think of the multiple choice example. E [ b p ] = E [ Y n ] = 1 n E [ Y ] = 1 n np = p Y n is an unbiased estimate. 5

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Var[ b p ] = E [ b p 2 ] - ( E [ b p ]) 2 = E Y 2 n 2 - E " Y n #! 2 = 1 n 2 Var Y = 1 n 2 np (1 - p ) = p (1 - p ) n Estimate Y n improves with n . 6
The Prosecutor’s Fallacy. Conditional Probability, Binomial Dis- tribution, and the Law This next example is well-known in the law. It was used by the Dream Team in the O.J. Simpson trial when a prosection witness was giving testimony about odds involving blood samples. In “People v. Collins , 68 Cal. 2d 319, 438 P.2d. 33 (1968) (en banc), an elderly woman who, while walking through an alley in the San Pedro area of Los Angeles, was assaulted from behind and robbed. The victim said that she managed to see a young woman with blond hair run from the scene. Another witness said that a Cau-

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## This note was uploaded on 01/03/2012 for the course EE 1244 taught by Professor Drera during the Fall '10 term at Conestoga.

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L9_rev2 - Binomial Distribution Assumptions n independent...

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