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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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).ThedistributionisXsXtXsHypergeometric(N;M;K (NK);x (N¡M¡k) N;M;KNegativeBinomial(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