Solutions

Solutions for Chapter 2: Statistical Model

The most general model is a model which takes into account different probabilities for defective parts at each day of the week. Such a model is given by a sample https://www.w3.org/1998/Math/MathML"> X = X 1 , … , X 7 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5263.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of independent r.v.’s with https://www.w3.org/1998/Math/MathML"> X i ∼ B i n ⁡ n , p i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5264.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where https://www.w3.org/1998/Math/MathML"> p i ∈ ( 0,1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5265.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and, since x ₆ and x ₇ describe the number of defective items at the weekends, https://www.w3.org/1998/Math/MathML"> p j > p i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5266.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for https://www.w3.org/1998/Math/MathML"> j = 6,7 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5267.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> i < 6 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5268.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> If one can assume that during the working days the production is of the same quality, and the probability of producing defective parts at the weekends is https://www.w3.org/1998/Math/MathML"> 50 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5269.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> higher, then we can assume the following model: https://www.w3.org/1998/Math/MathML"> P = B i n ⁡ ( n , θ ) ⊗ 5 ⊗ B i n ⁡ ( n , 1.5 θ ) ⊗ 2 : θ ∈ 0 , 2 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5270.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

(Here also other answers are possible.)

Let https://www.w3.org/1998/Math/MathML"> X i t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5271.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be the color of an arbitrary flower collected in region https://www.w3.org/1998/Math/MathML"> i = 1 , … , 5 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5272.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> on day https://www.w3.org/1998/Math/MathML"> t = 1,2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5273.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Define https://www.w3.org/1998/Math/MathML"> P X i t =   violet   = p i t , v ,   P X i t =   white   = p i t , w ,   P X i t =   pink   = p i t , p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5274.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

with https://www.w3.org/1998/Math/MathML"> p i t , p = 1 - p i t , v - p i t , w . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5275.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Note that we have for each pair https://www.w3.org/1998/Math/MathML"> ( i , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5276.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> a threepoint distribution.

Let https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5277.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be a r.v. drawn from the described mixture of normal distributions. Denote the event, that https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5278.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is drawn from the first part, by https://www.w3.org/1998/Math/MathML"> F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5279.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Then https://www.w3.org/1998/Math/MathML"> P θ ( X ≤ x )   = P θ ( X ≤ x , F ) + P θ ( X ≤ x , F ‾ )   = P θ ( X ≤ x ∣ F ) P θ ( F ) + P θ ( X ≤ x ∣ F ‾ ) P θ ( F ‾ )   = Φ x - μ 1 σ 1 π + Φ x - μ 2 σ 2 ( 1 - π ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5280.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

where https://www.w3.org/1998/Math/MathML"> π https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5281.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is the proportion of the first part in the population. Thus, the statistical model https://www.w3.org/1998/Math/MathML"> P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5282.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5283.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> consists of all distributions with distribution 232functions https://www.w3.org/1998/Math/MathML"> Φ ⋅ - μ 1 σ 1 π + Φ ⋅ - μ 2 σ 2 ( 1 - π ) : μ j ∈ R , σ j 2 ∈ R + , j = 1,2 , π ∈ ( 0,1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5284.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

With the same argumentation as in Example 2.18 we can show that uniform distributions do not form an exponential family. Uniform distributions have the density https://www.w3.org/1998/Math/MathML"> f ( x ; θ ) = 1 [ 0 , θ ] ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5285.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Take https://www.w3.org/1998/Math/MathML"> θ 1 < θ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5286.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . The set https://www.w3.org/1998/Math/MathML"> N = θ 1 , θ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5287.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a https://www.w3.org/1998/Math/MathML"> P θ 1 - https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5288.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> null set, but https://www.w3.org/1998/Math/MathML"> P θ 2 ( N ) > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5289.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Thus, the measures are not equivalent, which is a contradiction to being an exponential family.

https://www.w3.org/1998/Math/MathML"> N 0 , σ 2   θ = σ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5290.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Since https://www.w3.org/1998/Math/MathML"> f ( x ; θ ) = 1 2 π 1 θ e x p ⁡ - 1 2 θ x 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5291.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> this family forms a 1-parameter exponential family with https://www.w3.org/1998/Math/MathML"> A ( θ ) = ( 2 π θ ) - 1 2 ,   h ( x ) = 1 ,   ζ ( θ ) = - 1 2 θ     and   T ( x ) = x 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5292.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

https://www.w3.org/1998/Math/MathML"> N 1 , σ 2 , θ = σ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5293.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Since https://www.w3.org/1998/Math/MathML"> f ( x ; θ ) = 1 2 π 1 θ e x p ⁡ - 1 2 θ ( x - 1 ) 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5294.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> this family forms a 1-parameter exponential family with https://www.w3.org/1998/Math/MathML"> A ( θ ) = ( 2 π θ ) - 1 2 ,   h ( x ) = 1 ,   ζ ( θ ) = - 1 2 θ     and   T ( x ) = ( x - 1 ) 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5295.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

alternatively https://www.w3.org/1998/Math/MathML"> f ( x ; θ )   = 1 2 π θ e x p ⁡ - 1 2 θ x 2 - 2 x + 1   = 1 2 π θ e x p ⁡ - 1 2 θ e x p ⁡ - 1 2 θ x 2 - 2 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5296.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

thus https://www.w3.org/1998/Math/MathML"> A ( θ ) = ( 2 π θ ) - 1 2 e x p ⁡ - 1 2 θ , h ( x ) = 1 , ζ ( θ ) = - 1 2 θ   and   T ( x ) = x 2 - 2 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5297.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

https://www.w3.org/1998/Math/MathML"> N μ , σ 2 , θ = μ , σ 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5298.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Since https://www.w3.org/1998/Math/MathML"> f ( x ; θ ) = 1 2 π 1 θ 1 e x p ⁡ - 1 2 θ 1 x - θ 2 2 = A ( θ ) e x p ⁡ - 1 2 θ 1 x 2 + θ 2 θ 1 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5299.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

with https://www.w3.org/1998/Math/MathML"> A ( θ ) = ( 2 π θ ) - 1 2 e x p ⁡ - θ 2 2 2 θ 1 , h ( x ) = 1 ζ 1 ( θ ) = - 1 2 θ 1 ,   T 1 ( x ) = x 2 , ζ 2 ( θ ) = θ 2 θ 1 ,   T 2 ( x ) = x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5300.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

the normal distributions form a 2-parameter exponential family.

233 https://www.w3.org/1998/Math/MathML"> N ( μ , μ ) , θ = μ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5301.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Since https://www.w3.org/1998/Math/MathML"> f ( x ; θ ) = 1 2 π θ e x p ⁡ - 1 2 θ ( x - θ ) 2 = 1 2 π θ e x p ⁡ - 1 2 θ x 2 + x - θ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5302.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

these distributions form a 1-parameter exponential family with https://www.w3.org/1998/Math/MathML"> A ( θ ) = ( 2 π θ ) - 1 2 e x p ⁡ - θ 2 ,   h ( x ) = e x p ⁡ ( x ) ,   ζ 1 ( θ ) = - 1 2 θ ,   T 1 ( x ) = x 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5303.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

Poisson: Since https://www.w3.org/1998/Math/MathML"> p ( x ; λ ) = e x p ⁡ ( - λ ) λ x x ! = e x p ⁡ ( - λ ) e x p ⁡ ( x l n ⁡ λ ) 1 x !     for   x = 0,1 , … https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5304.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

we obtain that Poisson distributions form a 1-parameter exponential family with https://www.w3.org/1998/Math/MathML"> A ( λ ) = e x p ⁡ ( - λ ) , h ( x ) = 1 x ! ,   ζ ( λ ) = l n ⁡ λ ,     and   T ( x ) = x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5305.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

https://www.w3.org/1998/Math/MathML"> G e o ⁡ ( p ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5306.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> : Since https://www.w3.org/1998/Math/MathML"> p ( x ) = p ( 1 - p ) x = p e x p ⁡ ( x l n ⁡ ( 1 - p ) )     for   x = 0,1 , … https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5307.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

geometric distributions form a 1-parameter exponential family with https://www.w3.org/1998/Math/MathML"> A ( p ) = p , h ( x ) = 1 ,   ζ ( p ) = l n ⁡ ( 1 - p ) ,   T ( x ) = x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5308.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

The Rayleigh distribution, defined by the density https://www.w3.org/1998/Math/MathML"> f ( x ; α ) = 2 α x e x p ⁡ - x 2 α 1 [ 0 , ∞ ) ( x )   α > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5309.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

belongs to a 1-parameter exponential family with https://www.w3.org/1998/Math/MathML"> A ( α ) = 2 α , h ( x ) = x 1 [ 0 , ∞ ) ( x ) , ζ ( α ) = - 1 α   and   T ( x ) = x 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5310.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

As an example consider the triangle distribution: Tri https://www.w3.org/1998/Math/MathML"> ( 0 , θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5311.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , defined by https://www.w3.org/1998/Math/MathML"> f ( x ; θ ) = 2 θ 1 - 2 θ x - θ 2 1 [ 0 , θ ] ( x ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5312.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

As in Example 2.18 or Problem 4, we show that measures from this family are not pairwise equivalent. Consider the set https://www.w3.org/1998/Math/MathML"> N = 1 2 , 1 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5313.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> For the parameter https://www.w3.org/1998/Math/MathML"> θ = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5314.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> we obtain https://www.w3.org/1998/Math/MathML"> P 1 ( N ) = 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5315.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and for https://www.w3.org/1998/Math/MathML"> θ = 1 2 P 1 2 ( N ) = 0 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5316.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Thus https://www.w3.org/1998/Math/MathML"> N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5317.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is not a null-set for all parameters – this is a contradiction to the assumption that all measures in an exponential family are pairwise equivalent.

(This is only one of the possible answers.)

234The joint distribution of both samples is given by https://www.w3.org/1998/Math/MathML"> f x ; λ 1 , λ 2   = ∏ i = 1 n 1     λ 1 e x p ⁡ - λ 1 x 1 i ∏ j = 1 n 2     λ 2 e x p ⁡ - λ 2 x 2 j   = λ 1 n 1 e x p ⁡ - λ 1 ∑ i = 1 n 1     x 1 i λ 2 n 2 e x p ⁡ - λ 2 ∑ j = 1 n 2     x 2 j   = λ 1 n 1 λ 2 n 2 e x p ⁡ - λ 1 ∑ i = 1 n 1     x 1 i - λ 2 ∑ j = 1 n 2     x 2 j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5318.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

We obtain for https://www.w3.org/1998/Math/MathML"> θ = λ 1 , λ 2 :   A ( θ ) = λ 1 n 1 λ 2 n 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5319.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> ζ 1 ( θ ) = - λ 1 , ζ 2 ( θ ) = - λ 2 , T 1 ( x ) = ∑ i = 1 n 1   x 1 i , T 2 ( x ) = ∑ j = 1 n 2   x 2 j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5320.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

So we have a two-parameter exponential family.

Multinomial distribution: The calculation is analogous to Example 2.19. We have https://www.w3.org/1998/Math/MathML"> p x 1 , … , x m - 1 ; π = n x 1 , … , x m π 1 x 1 ⋯ π m x m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5321.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

with https://www.w3.org/1998/Math/MathML"> π m = ∑ i = 1 m - 1   π i , x m = n - ∑ i = 1 m - 1   x i . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5322.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Set https://www.w3.org/1998/Math/MathML"> h ( x ) = n x 1 , … , x m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5323.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , then https://www.w3.org/1998/Math/MathML"> p x 1 , … , x m - 1 ; π = = h ( x ) e x p ⁡ l n ⁡ π 1 x 1 … π m x m = h ( x ) e x p ⁡ ∑ i = 1 m     x i l n ⁡ π i = h ( x ) e x p ⁡ ∑ i = 1 m - 1     x i l n ⁡ π i + n - ∑ i = 1 m - 1     x i l n ⁡ 1 - ∑ i = 1 m - 1     π i = h ( x ) e x p ⁡ ∑ i = 1 m - 1     x i l n ⁡ π i - l n ⁡ 1 - ∑ i = 1 m - 1     π i + n l n ⁡ 1 - ∑ i = 1 m - 1     π i = h ( x ) 1 - ∑ i = 1 m - 1     π i n e x p ⁡ ∑ i = 1 m - 1     x i l n ⁡ π i 1 - ∑ j = 1 m - 1     π j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq5324.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>