# Table of Common Distributions taken from Statistical Inference by Casella and Berger Discrete Distrbutions distribution pmf mean variance mgfmoment Bernoulli pp pe Betabinomial n n n N PDF document - DocSlides

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Binomial np 1 1 n np np 1 1 pe Discrete Uniform 1 N 1 1 1 12 1 it Geometric 1 0 1 pe 1 Note 1 is negative binomial1 p The distribution is memoryless Xs Xt Xs Hypergeometric NMK 1 K KM KM 1 NMK Negative Binomial rp 1 0 1 1 1 1 1 Poisson 5752 ID: 23465

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### Presentations text content in Table of Common Distributions taken from Statistical Inference by Casella and Berger Discrete Distrbutions distribution pmf mean variance mgfmoment Bernoulli pp pe Betabinomial n n n N

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Table of Common Distributions taken from Statistical Inference by Casella and Berger Discrete Distrbutions distribution pmf mean variance mgf/moment Bernoulli( (1 =0 1; (0 1) pp (1 )(1 )+ pe Beta-binomial( n;˛; )( `( `( )`( `( )`( `( n n Notes: If is binomial ( n;P )and is beta( ˛; ), then is beta-binomial( n;˛; ). Binomial( n;p )( (1 =1 ;:::;n np np (1 ) [(1 )+ pe Discrete Uniform( =1 ;:::;N +1 +1)( 1) 12 =1 it Geometric( (1 (0 1) pe (1 Note: 1 is negative binomial(1 ;p ). The distribution is memoryless X>s X>t )= X>s ). Hypergeometric( N;M;K )( =1 ;:::;K KM KM )( 1) N;M;K> Negative Binomial( r;p )( (1 (0 1) (1 (1 (1 (1 Poisson(  e 1) Notes: If is gamma( ˛; ), is Poisson( ), and is an integer, then )= ).
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Continuous Distributions distribution pdf mean variance mgf/moment Beta( ˛; `( `( )`( (1 (0 1) ;˛;ˇ> +1) 1+ =1 =0 Cauchy( ; 1+( > 0 does not exist does not exist does not exist Notes: Special case of Students's with 1 degree of freedom. Also, if X;Y are iid (0 1), is Cauchy `( )2 x> ;p Np ;t< Notes: Gamma( 2). Double Exponential( ; > t t Exponential( ;> t ;t< Notes: Gamma(1 ; ). Memoryless. is Weibull. is Rayleigh. log is Gumbel. ; `( )`( 1+( x> ; 22( 4) ; EX `( +2 )`( `( )`( ;n< Notes: ; = = , where the s are independent. ; Gamma( ˛; `( x> ;˛;ˇ> ˇt ;t< Notes: Some special cases are exponential ( =1)and ; = 2). If ;Y is Maxwell. is inverted gamma. Logistic( ; 1+ ˇ> t `(1 + ˇt Notes: The cdf is ; )= 1+ Lognormal( ; (log x> ;> 2( EX n Normal( ; > t Pareto( ˛; +1 x>˛; ˛;ˇ > ;ˇ> 1) 2) ;ˇ> 2 does not exist `( +1 `( (1+ +1 ;> ;> EX `( +1 )`( `( ;n even Notes: ; Uniform( a;b ;a 12 bt at Notes: If =0 ;b = 1, this is special case of beta ( = 1). Weibull( —; x> ;—;ˇ> `(1 + `(1 + (1 + EX `(1 + Notes: The mgf only exists for 1.