NotesOct4 - . A success = drawing a type A object ....

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1 . Markov’s inequality For any nonnegative random variable X and a number a > 0 P ( X > a ) EX a .
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2 . Chebyshev’s inequality For any random variable X with mean μ , and number h > 0 P ( | X - μ | > h ) Var( X ) h 2 .
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3 . The Law of Large Numbers For any ± > 0 P ± ¯ X n - μ ± ± > ± ) 0 as n → ∞ . Review of the important discrete random variables Most of the important discrete random vari- ables are based on Bernoulli trials: a sequence of independent trials with probability p for suc- cess in each trial. The Binomial random variable counts the number of succeses in n trials. Notation : X Bin( n,p ). p X ( k ) = ² n k ³ p k (1 - p ) n - k , k = 0 , 1 ,...,n. A Bin(1 ,p ) random variable is called Bernoulli .
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The Negative Binomial random variable counts the number of trials until the n th success. Notation : X NB( n,p ). p X ( k ) = ± k - 1 n - 1 ² p n (1 - p ) k - n , k = n,n + 1 ,.... A NB(1 ,p ) random variable is called Geomet- ric .
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The Hypergeometric random variable counts the number of succeses in n trials, but the probability of success is different at every trial. In a finite population there are N objects of type A and M objects of type B . A sample of n N + M objects are drawn withour replacement
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Unformatted text preview: . A success = drawing a type A object . Probability mass function: p X ( k ) = N k M n-k N + M n , max( n-M, 0) k min( n,N ) . The Poisson random variable is the limiting case of the Binomial random variable as: n ; p 0; np > 0. Notation : X Poiss( ). p X ( k ) = e- k k ! , k = 0 , 1 , 2 ,.... Reproducing property of Poisson random variables Let X 1 ,X 2 ,...,X n be independent Poisson ran-dom variables with parameters 1 , 2 ,..., n . Then the sum Y = X 1 + X 2 + ... + X n has the Poisson distribution with parameter 1 + 2 + ... + n . Review of the important continuous random variables 1. Continuous models on a bounded interval The Uniform random variable takes values in a bounded interval ( a,b ), and has a constant density f X ( x ) = ( 1 b-a if a x b otherwise. Notation : X U( a,b )....
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This note was uploaded on 12/03/2010 for the course OR&IE 3500 taught by Professor Samorodnitsky during the Fall '10 term at Cornell University (Engineering School).

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NotesOct4 - . A success = drawing a type A object ....

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