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Table of Common Distributions taken from Statistical Inference by Casella and Berger Discrete Table of Common Distributions taken from Statistical Inference by Casella and Berger Discrete

Table of Common Distributions taken from Statistical Inference by Casella and Berger Discrete - PDF document

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Table of Common Distributions taken from Statistical Inference by Casella and Berger Discrete - PPT Presentation

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

Binomial

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TableofCommonDistributionstakenfromStatisticalInferencebyCasellaandBergerDiscreteDistrbutions distributionpmfmeanvariancemgf/moment )(1n;®;¯ )+®¡x+¯) +¯+n) +¯ Notes:Ifisbinomial(n;P)andisbeta(®;¯),thenisbeta-binomial(n;®;¯n;p;:::;nnpnp)[(1DiscreteUniform( N;x (N¡ 12 NPNi)pp)x¡1;p2 p1¡p p2 1isnegativebinomial(1).Thedistributionis�Xs�Xt�XsHypergeometric(N;M;K (NK);x (N¡M¡k) N;M;K�NegativeBinomial(r;p prp) p2³p Poisson( ¸¸eNotes:Ifisgamma(®;¯isPoisson( ),andisaninteger,then ContinuousDistributions distributionpdfmeanvariancemgf/moment ®;¯ ;®;¯� ®+¯ ®+¯)2(®+¯1kQk¡1r+r ®+¯+r´tk Cauchy(µ;¾ ¡µ 0doesnotexistdoesnotexistdoesnotexistNotes:SpecialcaseofStudents'swith1degreeoffreedom.Also,ifX;Yareiid isCauchy 2 2xp 2¡1e¡x 2;p³1 1¡2t´p ;t Notes:Gamma( DoubleExponential(¹;¾ 2¾e¡jx¡¹j ¾;¹2¾2e Exponential( µe¡x ;µ� ;t Notes:Gamma(1).Memoryless. Weibull. 2X Rayleigh. Gumbel. 2) 1 22 2)³º1 º2´º1 2xº1¡2 2 ¡1 º2)x¢º1+º2 2;º2 22( º2¡2)2º1+º2¡2 º1(º2¡�4=1 22¡2n 2) 1 22 2)³º2 ;n 2º1=Â2º1 ,wherethesareindependent.®;¯ )¯®x®¡1e¡x ;®;¯�®¯®¯ ;t Notes:Somespecialcasesareexponential(=1)and =2).If 3q X Maxwell. invertedgamma.¹;¯ ¯e¡x¡¹ ¯ h¡x¡¹ ¯i2;¹¼2¯2 ¡(1+ Notes:Thecdfis¹;¯ ¡x¡¹ ¹;¾ p 2 xe¡¡¹)2 ;¾� 2e+¾2)¡e2¹+¾2=en2¾2 ¹;¾ p 2¡(x¡¹)2 2¾2;e¾2t2 Pareto(®;¯ x�®;®;¯� ;¯� ;¯�2doesnotexist ) 2)1 p 2 º)º ;º� ;º� ¡n 2) p ¼ 2)ºn evena;b b¡ax·bb+a 2(b¡a)2 e Notes:If=1,thisisspecialcaseofbeta(=1).Weibull(°;¯ ¯x°¡1e¡x° ;°;¯� ¡(1+ °)¯2 ¡(1+ (1+ °)i=¯n ¡(1+ Notes:Themgfonlyexistsfor