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# brand - Stat 5101 Notes Brand Name Distributions Charles J...

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Stat 5101 Notes: Brand Name Distributions Charles J. Geyer January 16, 2012 Contents 1 Discrete Uniform Distribution 2 2 General Discrete Uniform Distribution 2 3 Uniform Distribution 3 4 General Uniform Distribution 3 5 Bernoulli Distribution 4 6 Binomial Distribution 5 7 Hypergeometric Distribution 6 8 Poisson Distribution 7 9 Geometric Distribution 8 10 Negative Binomial Distribution 9 11 Normal Distribution 10 12 Exponential Distribution 12 13 Gamma Distribution 12 14 Beta Distribution 14 15 Multinomial Distribution 15 1

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16 Bivariate Normal Distribution 18 17 Multivariate Normal Distribution 19 18 Chi-Square Distribution 21 19 Student’s t Distribution 22 20 Snedecor’s F Distribution 23 21 Cauchy Distribution 24 22 Laplace Distribution 25 1 Discrete Uniform Distribution Abbreviation DiscUnif( n ). Type Discrete. Rationale Equally likely outcomes. Sample Space The interval 1, 2, . . . , n of the integers. Probability Mass Function f ( x ) = 1 n , x = 1 , 2 , . . . , n Moments E ( X ) = n + 1 2 var( X ) = n 2 - 1 12 2 General Discrete Uniform Distribution Type Discrete. Sample Space Any finite set S . 2
Probability Mass Function f ( x ) = 1 n , x S, where n is the number of elements of S . 3 Uniform Distribution Abbreviation Unif( a, b ). Type Continuous. Rationale Continuous analog of the discrete uniform distribution. Parameters Real numbers a and b with a < b . Sample Space The interval ( a, b ) of the real numbers. Probability Density Function f ( x ) = 1 b - a , a < x < b Moments E ( X ) = a + b 2 var( X ) = ( b - a ) 2 12 Relation to Other Distributions Beta(1 , 1) = Unif(0 , 1). 4 General Uniform Distribution Type Continuous. Sample Space Any open set S in R n . 3

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Probability Density Function f ( x ) = 1 c , x S where c is the measure (length in one dimension, area in two, volume in three, etc.) of the set S . 5 Bernoulli Distribution Abbreviation Ber( p ). Type Discrete. Rationale Any zero-or-one-valued random variable. Parameter Real number 0 p 1. Sample Space The two-element set { 0 , 1 } . Probability Mass Function f ( x ) = ( p, x = 1 1 - p, x = 0 Moments E ( X ) = p var( X ) = p (1 - p ) Addition Rule If X 1 , . . . , X k are IID Ber( p ) random variables, then X 1 + · · · + X k is a Bin( k, p ) random variable. Degeneracy If p = 0 the distribution is concentrated at 0. If p = 1 the distribution is concentrated at 1. Relation to Other Distributions Ber( p ) = Bin(1 , p ). 4
6 Binomial Distribution Abbreviation Bin( n, p ). Type Discrete. Rationale Sum of n IID Bernoulli random variables. Parameters Real number 0 p 1. Integer n 1. Sample Space The interval 0, 1, . . . , n of the integers. Probability Mass Function f ( x ) = n x p x (1 - p ) n - x , x = 0 , 1 , . . . , n Moments E ( X ) = np var( X ) = np (1 - p ) Addition Rule If X 1 , . . . , X k are independent random variables, X i being Bin( n i , p ) distributed, then X 1 + · · · + X k is a Bin( n 1 + · · · + n k , p ) random variable. Normal Approximation If np and n (1 - p ) are both large, then Bin( n, p ) ≈ N ( np, np (1 - p ) ) Poisson Approximation If n is large but np is small, then Bin( n, p ) Poi( np ) Theorem The fact that the probability mass function sums to one is equivalent to the binomial theorem: for any real numbers a and b n X k =0 n k a k b n - k = ( a + b ) n .

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