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NotesOct4

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

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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 .
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 ) = n k 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 p n (1 - p ) k - n , k = n, n + 1 , . . . . A NB(1 , p ) random variable is called Geomet- ric .
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 ²± M n-k ² ± N + M n ² , 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 → λ > 0. Notation : X ∼ Poiss( λ ). p X ( k ) = e-λ λ 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 λ 1 ,λ 2 ,...,λ n . Then the sum Y = X 1 + X 2 + ... + X n has the Poisson distribution with parameter λ 1 + λ 2 + ... + λ 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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NotesOct4 - A success = drawing a type A object Probability...

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