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Unformatted text preview: 3. Distributions Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 11/28/10 Goldsman 11/28/10 1 / 71 Outline 1 Discrete Distributions Bernoulli and Binomial Distributions Hypergeometric Distribution Geometric and Negative Binomial Distributions Poisson Distribution 2 Continuous Distributions Uniform Distribution Exponential, Erlang, and Gamma Distributions Other Continuous Distributions 3 Normal Distribution Basics Standard Normal Distribution Sample Mean of Normal Observations Central Limit Theorem Extensions of the Normal Distribution 4 Computer Stuff Goldsman 11/28/10 2 / 71 Discrete Distributions Bernoulli and Binomial Distributions Bernoulli( p ) and Binomial( n,p ) Distributions The Bern( p ) distribution is given by X = ( 1 w.p. p (“success”) w.p. q (“failure”) Recall: E[ X ] = p , Var( X ) = pq , and M X ( t ) = pe t + q . Further, X 1 ,...,X n iid ∼ Bern ( p ) ⇒ Y ≡ ∑ n i =1 X i ∼ Bin ( n,p ) . P ( Y = k ) = n k p k q n k , k = 0 , 1 ,...,n. Example: Toss 2 dice 5 times. Let Y be the number of 7’s you see. Y ∼ Bin (5 , 1 / 6) . Then, e.g., P ( Y = 4) = 5 4 1 6 4 5 6 5 4 . Goldsman 11/28/10 3 / 71 Discrete Distributions Bernoulli and Binomial Distributions Y ∼ Bin ( n,p ) implies E[ Y ] = E n X i =1 X i = n X i =1 E[ X i ] = np and, similarly, Var( Y ) = npq. We’ve already seen that M Y ( t ) = ( pe t + q ) n . Binomials add up: If Y 1 ,...,Y k are indep and Y i ∼ Bin ( n i ,p ) , then k X i =1 Y i ∼ Bin k X i =1 n i ,p . Goldsman 11/28/10 4 / 71 Discrete Distributions Hypergeometric Distribution Hypergeometric Distribution You have a objects of type 1 and b objects of type 2. Select n objects w/o replacement from the a + b . Let X be the number of type 1’s selected. P ( X = k ) = ( a k )( b n k ) ( a + b n ) , k = 0 , 1 ,...,n. Goldsman 11/28/10 5 / 71 Discrete Distributions Hypergeometric Distribution After some algebra, it turns out that E[ X ] = n a a + b and Var( X ) = n a a + b 1 a a + b a + b n a + b 1 . Example: 25 sox in a box. 15 red, 10 blue. Pick 7 w/o replacement. P ( exactly 3 reds are picked ) = ( 15 3 )( 10 4 ) ( 25 7 ) Goldsman 11/28/10 6 / 71 Discrete Distributions Geometric and Negative Binomial Distributions Geometric( p ) and Negative Binomial( r,p ) Distributions Definition: Suppose we consider an infinite sequence of indep Bern( p ) trials. Let Z equal the number of trials until the first success is obtained. The event Z = k corresponds to k 1 failures, and then a success. Thus, P ( Z = k ) = q k 1 p, k = 1 , 2 ,..., and we say that Z has the Geometric( p ) distribution . Goldsman 11/28/10 7 / 71 Discrete Distributions Geometric and Negative Binomial Distributions The mgf of the Geom( p ) is M Z ( t ) = E[ e tZ ] = ∞ X k =1 e tk q k 1 p = pe t ∞ X k =0 ( qe t ) k = pe t 1 qe t , for qe t < 1 ....
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This note was uploaded on 08/27/2011 for the course ISYE 3770 taught by Professor Goldsman during the Spring '07 term at Georgia Tech.
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