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# 3.05 - Chapter3,Section5 John J Currano 1 Formulas Related...

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1 Chapter 3, Section 5 Geometric Distributions John J Currano, 02/25/2010

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2 Formulas Related to the Geometric Series ) 1 ( 1 0 x x k k - = = (1) (2) (3) (4) (5) Geometric Series (provided | x| < 1) from differentiating (1) multiply (2) by x to get (3); then differentiate (3) to get (4) from multiplying (4) by x 2 1 1 ) 1 ( 1 x kx k k - = = - 2 0 ) 1 ( x x kx k k - = = 3 4 2 1 1 2 ) 1 ( 1 ) 1 ( )] 1 )( 1 ( 2 [ ) 1 ( 1 x x x x x x x k k k - + = - - - - - = = - 3 0 2 ) 1 ( ) 1 ( x x x x k k k - + = =
3 Consider again a binomial experiment : Each trial results in one of two outcomes: success, S , and failure, F. P ( S ) = p and P ( F ) = q = (1 - p ); remain the same from trial to trial . The trials are independent. But now, do not specify the number of trials in advance, but Repeat the experiment until we obtain the first success. Let Y be the waiting time until the first success : the number of the trial on which the first success occurs, or the number of trials up to, and including, the first success. Warning : some let Y be the number of trials (failures) before the first success. Then Y is said to have a geometric distribution with parameter p provided 0 < p < 1. We write Y ~ Geometric( p ) .

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4 Consider again a binomial experiment : Independent trials resulting in either success, S , or failure, F. P ( S ) = p and P ( F ) = q = (1 - p ); remain the same from trial to trial .
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3.05 - Chapter3,Section5 John J Currano 1 Formulas Related...

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