Special Results

Question 1

Xi ~ Bi(mi, p) ∀i = 1,2,…,n

Answer 1

Sn ~ Bi(∑mi, p)

Sn ~ Bi(nm, p) if mi = m ∀i

Question 2

Xi ~ NB(ri, p) ∀i = 1,2,…,n

Answer 2

Sn ~ NB(∑ri, p)

Sn ~ NB(nr, p) if ri = r ∀i

Question 3

Xi ~ Po(λi) ∀i = 1,2,…,n

Answer 3

Sn ~ Po(∑λi)

Sn ~ Po(nλ) if λi = λ ∀i

Question 4

Xi ~ Ga(ri, λ) ∀i = 1,2,…,n

Answer 4

Sn ~ Ga(∑ri, λ)

Sn ~ Ga(nr, λ) if ri = r ∀i

Question 5

Xi ~ N(μi, σi²) ∀i = 1,2,…,n

Answer 5

Sn ~ N(∑aiμi, ∑ai²σi²)

Sn ~ N(nμ, nσ²) if μi = μ, σi² = σ ∀i

Question 6

Xi ~ N(μi, σi²) ∀i = 1,2,…,n

reproductive family

Answer 6

∑aiXi ~ N(∑aiμi, ∑ai²σi²), a is a constant

Question 7

Xi ~ Exp(λ) ∀i = 1,2,…,n

Answer 7

Sn ~ Ga(n,λ)

Question 9

Xi ~ Ga(1, λ) ∀i = 1,2,…,n

Answer 9

Sn ~ Ga(n,λ)

Question 10

X ~ N(μ, σ²)

standardizing

Answer 10

[X - N]/σ ~ N(0,1)