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l16_negbino - 04/01/2009 CS206 - Intro. to Discrete...

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Unformatted text preview: 04/01/2009 CS206 - Intro. to Discrete Structures II 1 Geometric & Negative Binomial Distributions Reading: Ross, Ch 4., Sec. 8.1 & 8.2. Wednesday, April 1, 2009 04/01/2009 CS206 - Intro. to Discrete Structures II 2 Example 1 Independent trials, consisting of the flipping of a coin with P(head) = .8, are repeated until a head occurs. What is the probability of having to flip the coin k times? Let X be the number of times the coin is flipped. P{X=k} = 0.2 k-1 0.8, k=1,2, Note: range of X is infinite (set of all positive integers) k-1 tails, then k-th has to be head Wednesday, April 1, 2009 04/01/2009 CS206 - Intro. to Discrete Structures II 3 Geometric Random Variable #trials until head (success occurs), when all trials are independent Bernoulli, has geometric distribution : P{X=k} = p X (k) = (1-p) k-1 p, k=1,2, Geometric distribution has one parameter, p probability of success of each individual trial. X ~ Ge(p). Wednesday, April 1, 2009 04/01/2009 CS206 - Intro. to Discrete Structures II 4 Example 2 p X (x) x p=0.6 Wednesday, April 1, 2009 04/01/2009 CS206 - Intro. to Discrete Structures II 5 CDF of Geometric RV Cumulative distribution function (CDF) of a geometric rv: P(x) = P{X k} = l=1 k p X (l) = l=1 k (1-p) l-1 p = p l=1 k (1-p) l-1 = p [ 1 (1-p) k ] / [ 1-(1-p)] = 1 (1-p) k , k =1,2, Another way: P{X k} = 1 P{X>k} = 1 P(exactly k failures) = 1- (1-p) k Wednesday, April 1, 2009 04/01/2009 CS206 - Intro. to Discrete Structures II 6 Example 2 P(x) x p=0.6 Wednesday, April 1, 2009 04/01/2009 CS206 - Intro. to Discrete Structures IICS206 - Intro....
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This note was uploaded on 03/24/2011 for the course CS 206 taught by Professor Fredman during the Spring '08 term at Rutgers.

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l16_negbino - 04/01/2009 CS206 - Intro. to Discrete...

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