67((/ )5$0,1* (1*(1+$5,$ - Ig...
127
Transcript of 67((/ )5$0,1* (1*(1+$5,$ - Ig...
• • • • • • • • • • •
σ
≅
σ
σ σ
λ
σ
λ
λ λ
= λ
λ
ψ ≥
λ λ ≤
ψ
λ λ ≤
λ
λ ≤
σ
σ
σ
≥
≤
≤
σ ρ ρ
σ ρ ρ
σ
θ
λ ≤
λ
λ
≤
≤
≤ ≤ θ ≤ ≤
θ
≤
≤
λ ≥
λ
≤
σ σ
σ
σ
ρ γ γ
ρ
β α λ λ
α
ααα
λ
σ ρ
σ ρ
λ
λ ≤σ
π
=
π
=
⎥ ⎦
⎤ ⎢ ⎣
⎡ +
π =
π
=
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡
+ −
− − −
+ =
λ
λ γ
≤ λ ≤
λ γ
γ
λ
λ σ
λ
γ γ
σ
σ
σ
σσ
σ
ρ γ γ
σ ρ
ρ
• λ ≤ ρ
• λ ρ λ
• λ ρ λ
λ
•
•
•
+ + +
=
γ γ
• λ λ
• λ ≥λ
λ
λ σ
σ
σ
•
•
•
•
≤
π π
•
•
•
γ γ
γ γ
γ
γ
γ
γ
γ
γ γ
γ γ
γ γ
φ
φ
≥
≥
≤
≥
σ
λ
λ ≥ ⇒
λ
≤
≤
λ
≤
≤
λ
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ ≤
⎟ ⎠
⎞ ⎜ ⎝
⎛ ≤
σ
σ
Ψ
Ψ σ σ
Ψ Ψ
λ
λ
≤
≤
Ψ −
+
∑
∴
ρ γ γ
σ ρ
( ) ⎪ ⎭
⎪ ⎬ ⎫
⎪ ⎩
⎪ ⎨ ⎧
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ +
π
( ) ⎪ ⎭
⎪ ⎬ ⎫
⎪ ⎩
⎪ ⎨ ⎧
⎥ ⎦
⎤ ⎢ ⎣
⎡
× × ×
+ × π
σ
λ ρ λ
λ
= ρ
ρ
λ λ
λ ≥ ⇒
≤
≤
≤
≤
λ
≤
≤
σ ( ) −
σ ( ) −
Ψ
σ σ −
Ψ Ψ
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ =
σ
λ
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠
⎞ ⎜ ⎝
⎛
λ
λ
≤
≤
∑
∴
σ
σ α α α α
β
β
β
β
β
β
β β
β
η π
η
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛
+ + −
+ =
σ φ
= φ
σ
σ
λ σ
λ
λ λ
γ γ
ρ γ γ
σ ρ
•
•
•
•
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎣
⎡
+
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ −
− −
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ −
+ =
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎣
⎡
+ ⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠
⎞ ⎜ ⎝
⎛ − − × × ×
− −
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠
⎞ ⎜ ⎝
⎛ − − ×
+ =
⇒
λ ×
λ
α
α
β ( ) [ ] λ λ α + − + ×
β ρ
ρ ( ) [ ] λ β β − +
ρ
ρ σ =
ψ σ σ
λ
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ −
λ ≤
⎟ ⎠
⎞ ⎜ ⎝
⎛ − ≤
λ
λ
λ ⇒
λ
λ
⎟ ⎠ ⎞
⎜ ⎝ ⎛
≤
≤
λ
λ
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ ≤
⎟ ⎠
⎞ ⎜ ⎝
⎛ ≤
γ ρ
×
ρ
σ ρ
ρ σ =
λ
≤
ψ σ σ
≤
λ
λ
λ ≥ ⇒
λ
λ
≤
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛
≤
λ
≤
≤
λ
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ ≤
⎟ ⎠
⎞ ⎜ ⎝
⎛ ≤
λ
λ
λ ≥ ⇒ ⇒
λ
≤
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ ⎟ ⎠
⎞ ⎜ ⎝
⎛
≤
λ
≤
≤
λ
≤
≤
⎟ ⎠ ⎞
⎜ ⎝ ⎛
λ
σ ( ) −
σ ( ) −
Ψ
σ σ
Ψ σ σ
Ψ Ψ
λ
≤
Ψ −
=
∴
ρ γ γ
σ ρ
σ
λ
ρ λ
λ
= ρ
ρ
= × = = ρ σ
λ
λ ≥ ⇒
λ
λ
⎟ ⎠ ⎞
⎜ ⎝ ⎛
≤
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛
≤
λ
≤
⎟ ⎠
⎞ ⎜ ⎝
⎛ − ≤
λ
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ ≤
≤
σ
σ
Ψ
σ σ
Ψ Ψ
λ
≤
≤
∴
ρ
( ) π
( ) π
( ) π
( )
× × × π
ρ γ γ
σ ρ
•
•
( ) π
( )
× × × π
•
( ) ⎪ ⎭
⎪ ⎬ ⎫
⎪ ⎩
⎪ ⎨ ⎧
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ +
π
( ) ⎪ ⎭
⎪ ⎬ ⎫
⎪ ⎩
⎪ ⎨ ⎧
⎥ ⎦
⎤ ⎢ ⎣
⎡
× × ×
+ × π
•
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎣
⎡
+
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ −
− −
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ −
+ =
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎣
⎡
+ ⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠
⎞ ⎜ ⎝
⎛ − − × × ×
− −
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠
⎞ ⎜ ⎝
⎛ − − ×
+ =
⇒
λ ×
λ
α
α
β ( ) [ ] λ λ α + − + ×
β ρ
ρ ( ) [ ] λ β β − +
ρ
ρ σ =
ψ σ σ
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠ ⎞
⎜ ⎝ ⎛
=
σ
λ
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠
⎞ ⎜ ⎝
⎛
λ
λ ⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ − −
=
≤
⎥ ⎦
⎤ ⎢ ⎣
⎡ × − − ×
= ≤
λ
λ
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠ ⎞
⎜ ⎝ ⎛
σ
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠
⎞ ⎜ ⎝
⎛
λ ⇒
λ
⎟ ⎠ ⎞
⎜ ⎝ ⎛
≤
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ ⎟ ⎠
⎞ ⎜ ⎝
⎛
≤
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠ ⎞
⎜ ⎝ ⎛
=
σ
λ
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠
⎞ ⎜ ⎝
⎛ ×
λ
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠ ⎞
⎜ ⎝ ⎛
=
σ
λ
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ ⎟ ⎠
⎞ ⎜ ⎝
⎛
λ
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ ≤
⎟ ⎠
⎞ ⎜ ⎝
⎛ ≤
γ ρ
>
• •
• •
• • • •
• •
•
•
•
•
•
γ γ
γ γ
•
•
( ) Σ + − =
≤
•
•
•
![1 Appendix: Common distributionsfaculty.chicagobooth.edu/nicholas.polson/teaching/41900/Appendices...1 Appendix: Common distributions ... Beta • A random variable X ∈ [0,1] has](https://static.fdocument.org/doc/165x107/5ae3e1407f8b9a595d8f03f5/1-appendix-common-appendix-common-distributions-beta-a-random-variable.jpg)
