Stat 5101 Lecture Notes - School of Statistics

228 Stat 5101 (Geyer) Course NotesPoissonIf X 1 , ..., X k are independent with X i ∼ Poi(µ i ), thenX 1 + ···+X k ∼Poi(µ 1 + ···+µ k ).(C.5)ExponentialIf X 1 , ..., X k are i. i. d. Exp(λ), thenX 1 + ···+X k ∼Gam(n, λ).(C.6)• All the exponential distributions must have the same rate parameter λ.GammaIf X 1 , ..., X k are independent with X i ∼ Gam(α i ,λ), thenX 1 + ···+X k ∼Gam(α 1 + ···+α k ,λ).(C.7)• All the gamma distributions must have the same rate parameter λ.• (C.6) is the special case of (C.7) obtained by setting α 1 = ···=α k =1.Chi-SquareIf X 1 , ..., X k are independent with X i ∼ chi 2 (n i ), thenX 1 + ···+X k ∼chi 2 (n 1 + ···+n k ).(C.8)• (C.8) is the special case of (C.7) obtained by settingα i = n i /2 and λ i =1/2, i =1,...,k.NormalIf X 1 , ..., X k are independent with X i ∼N(µ i ,σi 2 ), thenX 1 + ···+X k ∼N(µ 1 +···+µ k ,σ 2 1 +···+σ 2 k).(C.9)Linear Combination of Normals If X 1 , ...,X k are independent with X i ∼N (µ i ,σi 2) and a 1, ..., a k are constants, then(k∑k∑a i X i ∼N a i µ i ,i=1i=1)k∑a 2 i σi2 . (C.10)• (C.9) is the special case of (C.10) obtained by setting a 1 = ···=a k =1.i=1CauchyIf X 1 , ..., X k are independent with X i ∼ Cauchy(µ, σ), thenX 1 + ···+X k ∼Cauchy(nµ, nσ).(C.11)