피어슨 정리 증명
📂확률분포론 피어슨 정리 증명 정리 ∑ i = 1 k N i = n & ∑ i = 1 k p i = 1
\sum_{i=1}^{k} N_{i} = n \qquad \& \qquad \sum_{i=1}^{k} p_{i} = 1
i = 1 ∑ k N i = n & i = 1 ∑ k p i = 1 p = ( p 1 , ⋯ , p k ) ∈ [ 0 , 1 ] k \mathbf{p} = \left( p_{1} , \cdots , p_{k} \right) \in [0,1]^{k} p = ( p 1 , ⋯ , p k ) ∈ [ 0 , 1 ] k n ∈ N n \in \mathbb{N} n ∈ N k ∈ N k \in \mathbb{N} k ∈ N 범주 에 대해 랜덤벡터 ( N 1 , ⋯ , N k ) \left( N_{1} , \cdots , N_{k} \right) ( N 1 , ⋯ , N k ) 다항분포 M k ( n ; p ) M_{k} \left( n ; \mathbf{p} \right) M k ( n ; p ) n → ∞ n \to \infty n → ∞ 통계량 S S S 카이제곱분포 χ 2 ( k − 1 ) \chi^{2} \left( k - 1 \right) χ 2 ( k − 1 ) 분포수렴 한다.
S : = ∑ j = 1 k ( N j − n p j ) 2 n p j → D χ 2 ( k − 1 )
S := \sum_{j=1}^{k} {{ \left( N_{j} - n p_{j} \right)^{2} } \over { n p_{j} }} \overset{D}{\to} \chi^{2} \left( k-1 \right)
S := j = 1 ∑ k n p j ( N j − n p j ) 2 → D χ 2 ( k − 1 )
[ 0 , 1 ] k = [ 0 , 1 ] × ⋯ × [ 0 , 1 ] [0,1]^{k} = [0,1] \times \cdots \times [0,1] [ 0 , 1 ] k = [ 0 , 1 ] × ⋯ × [ 0 , 1 ] k k k → D \overset{D}{\to} → D 분포수렴 을 의미한다.χ 2 ( r ) \chi^{2} \left( r \right) χ 2 ( r ) 자유도 r r r 카이제곱분포 를 의미한다.설명 사실 피어슨 정리 라는 표현이 자주 쓰이는 것은 아니다. 솔직히 한 번 밖에 못 봤는데통계량
X 2 : = ∑ j = 1 k ( O j − E j ) 2 E j
\mathcal{X}^{2} := \sum_{j=1}^{k} {{ \left( O_{j} - E_{j} \right)^{2} } \over { E_{j} }}
X 2 := j = 1 ∑ k E j ( O j − E j ) 2 피어슨 카이제곱 통계량 pearson Chi-squared statistic 이라 부르고 이것이 카이제곱분포로 분포수렴한다는 식의 스테이트먼트로만 쓰이기 때문이다. 정확히 정리라는 인식을 가지고 공부하기보단 냅다 실용적인 가설검정 이야기로 넘어가기 때문에 엄밀한 수리적 증명을 경시해서 그렇듯 하다.
증명 p \mathbf{p} p p j > 0 p_{j} > 0 p j > 0 S : = ∑ j = 1 k ( X j − n p j ) 2 n p j
S := \sum_{j=1}^{k} {{ \left( X_{j} - n p_{j} \right)^{2} } \over { n p_{j} }}
S := j = 1 ∑ k n p j ( X j − n p j ) 2 ∑ j = 1 k ( N j − n p j ) = 0 \sum_{j=1}^{k} \left( N_{j} - n p_{j} \right) = 0 ∑ j = 1 k ( N j − n p j ) = 0 k k k S = ∑ j = 1 k ( X j − n p j ) 2 n p j = ∑ j = 1 k − 1 ( X j − n p j ) 2 n p j + ( X k − n p k ) 2 n p k = ∑ j = 1 k − 1 ( X j − n p j ) 2 n p j + ( ∑ j = 1 k − 1 ( X j − n p j ) ) 2 n p k
\begin{align*}
S =& \sum_{j=1}^{k} {{ \left( X_{j} - n p_{j} \right)^{2} } \over { n p_{j} }}
\\ =& \sum_{j=1}^{k-1} {{ \left( X_{j} - n p_{j} \right)^{2} } \over { n p_{j} }} + {{ \left( X_{k} - n p_{k} \right)^{2} } \over { n p_{k} }}
\\ =& \sum_{j=1}^{k-1} {{ \left( X_{j} - n p_{j} \right)^{2} } \over { n p_{j} }} + {{ \left( \sum_{j=1}^{k-1} \left( X_{j} - n p_{j} \right) \right)^{2} } \over { n p_{k} }}
\end{align*}
S = = = j = 1 ∑ k n p j ( X j − n p j ) 2 j = 1 ∑ k − 1 n p j ( X j − n p j ) 2 + n p k ( X k − n p k ) 2 j = 1 ∑ k − 1 n p j ( X j − n p j ) 2 + n p k ( ∑ j = 1 k − 1 ( X j − n p j ) ) 2 N k N_{k} N k k k k N ~ : = ( N 1 , ⋯ , N k − 1 ) \widetilde{N} := \left( N_{1} , \cdots , N_{k-1} \right) N := ( N 1 , ⋯ , N k − 1 ) p ~ : = ( p 1 , ⋯ , p k − 1 ) \widetilde{\mathbf{p}} := \left( p_{1} , \cdots , p_{k-1} \right) p := ( p 1 , ⋯ , p k − 1 ) N ~ \widetilde{N} N 공분산행렬 Σ ~ \widetilde{\Sigma} Σ
다항분포의 공분산행렬 : 랜덤벡터 X : = ( X 1 , ⋯ , X k ) \mathbf{X} := \left( X_{1} , \cdots , X_{k} \right) X := ( X 1 , ⋯ , X k ) 다항분포 M k ( n , p ) M_{k} \left( n, \mathbf{p} \right) M k ( n , p ) 공분산행렬 은 다음과 같다.
Cov ( X ) = n [ p 1 ( 1 − p 1 ) − p 1 p 2 ⋯ − p 1 p k − p 2 p 1 p 2 ( 1 − p 2 ) ⋯ − p 2 p 2 ⋮ ⋮ ⋱ ⋮ − p k p 1 − p k p 2 ⋯ p k ( 1 − p k ) ]
\operatorname{Cov} \left( \mathbf{X} \right) = n \begin{bmatrix}
p_{1} \left( 1 - p_{1} \right) & - p_{1} p_{2} & \cdots & - p_{1} p_{k}
\\ - p_{2} p_{1} & p_{2} \left( 1 - p_{2} \right) & \cdots & - p_{2} p_{2}
\\ \vdots & \vdots & \ddots & \vdots
\\ - p_{k} p_{1} & - p_{k} p_{2} & \cdots & p_{k} \left( 1 - p_{k} \right)
\end{bmatrix}
Cov ( X ) = n p 1 ( 1 − p 1 ) − p 2 p 1 ⋮ − p k p 1 − p 1 p 2 p 2 ( 1 − p 2 ) ⋮ − p k p 2 ⋯ ⋯ ⋱ ⋯ − p 1 p k − p 2 p 2 ⋮ p k ( 1 − p k )
k k k Cov ( N i , N j ) \operatorname{Cov} \left( N_{i} , N_{j} \right) Cov ( N i , N j ) 항등행렬 I k − 1 I_{k-1} I k − 1 1 n Σ ~ = 1 n Cov ( N ~ ) = [ p 1 ( 1 − p 1 ) − p 1 p 2 ⋯ − p 1 p k − 1 − p 2 p 1 p 2 ( 1 − p 2 ) ⋯ − p 2 p 2 ⋮ ⋮ ⋱ ⋮ − p k − 1 p 1 − p k − 1 p 2 ⋯ p k − 1 ( 1 − p k − 1 ) ] = [ p 1 0 ⋯ 0 0 p 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ p k − 1 ] − [ − p 1 2 − p 1 p 2 ⋯ − p 1 p k − 1 − p 2 p 1 − p 2 2 ⋯ − p 2 p 2 ⋮ ⋮ ⋱ ⋮ − p k − 1 p 1 − p k − 1 p 2 ⋯ − p k − 1 2 ] = [ p 1 0 ⋯ 0 0 p 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ p k − 1 ] − [ p 1 p 2 ⋮ p k − 1 ] [ p 1 p 2 ⋯ p k − 1 ] = [ p 1 0 ⋯ 0 0 p 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ p k − 1 ] − p ~ p ~ T = I k − 1 p ~ − p ~ p ~ T
\begin{align*}
& {{ 1 } \over { n }} \widetilde{\Sigma}
\\ =& {{ 1 } \over { n }} \operatorname{Cov} \left( \widetilde{N} \right)
\\ =& \begin{bmatrix}
p_{1} \left( 1 - p_{1} \right) & - p_{1} p_{2} & \cdots & - p_{1} p_{k-1}
\\ - p_{2} p_{1} & p_{2} \left( 1 - p_{2} \right) & \cdots & - p_{2} p_{2}
\\ \vdots & \vdots & \ddots & \vdots
\\ - p_{k-1} p_{1} & - p_{k-1} p_{2} & \cdots & p_{k-1} \left( 1 - p_{k-1} \right)
\end{bmatrix}
\\ =& \begin{bmatrix}
p_{1} & 0 & \cdots & 0
\\ 0 & p_{2} & \cdots & 0
\\ \vdots & \vdots & \ddots & \vdots
\\ 0 & 0 & \cdots & p_{k-1}
\end{bmatrix} - \begin{bmatrix}
- p_{1}^{2} & - p_{1} p_{2} & \cdots & - p_{1} p_{k-1}
\\ - p_{2} p_{1} & - p_{2}^{2} & \cdots & - p_{2} p_{2}
\\ \vdots & \vdots & \ddots & \vdots
\\ - p_{k-1} p_{1} & - p_{k-1} p_{2} & \cdots & - p_{k-1}^{2} \end{bmatrix}
\\ =& \begin{bmatrix}
p_{1} & 0 & \cdots & 0
\\ 0 & p_{2} & \cdots & 0
\\ \vdots & \vdots & \ddots & \vdots
\\ 0 & 0 & \cdots & p_{k-1}
\end{bmatrix} - \begin{bmatrix} p_{1} \\ p_{2} \\ \vdots \\ p_{k-1} \end{bmatrix} \begin{bmatrix} p_{1} & p_{2} & \cdots & p_{k-1} \end{bmatrix}
\\ =& \begin{bmatrix}
p_{1} & 0 & \cdots & 0
\\ 0 & p_{2} & \cdots & 0
\\ \vdots & \vdots & \ddots & \vdots
\\ 0 & 0 & \cdots & p_{k-1}
\end{bmatrix} - \widetilde{\mathbf{p}} \widetilde{\mathbf{p}}^{T}
\\ =& I_{k-1} \widetilde{\mathbf{p}} - \widetilde{\mathbf{p}} \widetilde{\mathbf{p}}^{T}
\end{align*}
= = = = = = n 1 Σ n 1 Cov ( N ) p 1 ( 1 − p 1 ) − p 2 p 1 ⋮ − p k − 1 p 1 − p 1 p 2 p 2 ( 1 − p 2 ) ⋮ − p k − 1 p 2 ⋯ ⋯ ⋱ ⋯ − p 1 p k − 1 − p 2 p 2 ⋮ p k − 1 ( 1 − p k − 1 ) p 1 0 ⋮ 0 0 p 2 ⋮ 0 ⋯ ⋯ ⋱ ⋯ 0 0 ⋮ p k − 1 − − p 1 2 − p 2 p 1 ⋮ − p k − 1 p 1 − p 1 p 2 − p 2 2 ⋮ − p k − 1 p 2 ⋯ ⋯ ⋱ ⋯ − p 1 p k − 1 − p 2 p 2 ⋮ − p k − 1 2 p 1 0 ⋮ 0 0 p 2 ⋮ 0 ⋯ ⋯ ⋱ ⋯ 0 0 ⋮ p k − 1 − p 1 p 2 ⋮ p k − 1 [ p 1 p 2 ⋯ p k − 1 ] p 1 0 ⋮ 0 0 p 2 ⋮ 0 ⋯ ⋯ ⋱ ⋯ 0 0 ⋮ p k − 1 − p p T I k − 1 p − p p T p ~ T \widetilde{\mathbf{p}}^{T} p T p ~ \widetilde{\mathbf{p}} p 전치행렬 을 의미한다. 이제 P ~ : = I k − 1 p ~ \widetilde{P} := I_{k-1} \widetilde{\mathbf{p}} P := I k − 1 p
셔먼-모리슨 공식 : ( A + u v T ) − 1 \left( A + \mathbf{u} \mathbf{v}^{T} \right)^{-1} ( A + u v T ) − 1 ( A + u v T ) − 1 = A − 1 − A − 1 u v T A − 1 1 + v T A − 1 u
\left( A + \mathbf{u} \mathbf{v}^{T} \right)^{-1} = A^{-1} - {{ A^{-1} \mathbf{u} \mathbf{v}^{T} A^{-1} } \over { 1 + \mathbf{v}^{T} A^{-1} \mathbf{u} }}
( A + u v T ) − 1 = A − 1 − 1 + v T A − 1 u A − 1 u v T A − 1
1 n Σ ~ {{ 1 } \over { n }} \widetilde{\Sigma} n 1 Σ 역행렬 을 계산해보면 대각행렬
P ~ = I k − 1 p ~ = diag ( p 1 , ⋯ , p k − 1 )
\widetilde{P} = I_{k-1} \widetilde{\mathbf{p}} = \text{diag} \left( p_{1} , \cdots , p_{k-1} \right)
P = I k − 1 p = diag ( p 1 , ⋯ , p k − 1 ) 대각행렬 P ~ − 1 = diag ( p 1 − 1 , ⋯ , p k − 1 − 1 ) \widetilde{P}^{-1} = \text{diag} \left( p_{1}^{-1} , \cdots , p_{k-1}^{-1} \right) P − 1 = diag ( p 1 − 1 , ⋯ , p k − 1 − 1 ) ( 1 n Σ ~ ) − 1 = ( P ~ + ( − p ~ p ~ T ) ) − 1 = P ~ − 1 + P ~ − 1 p ~ p ~ T P ~ − 1 1 − p ~ T P ~ − 1 p ~ = P ~ − 1 + I k − 1 I k − 1 1 − I k − 1 p ~ = P ~ − 1 + I k − 1 I k − 1 1 − I k − 1 p ~ = P ~ − 1 + 1 1 − p 1 − ⋯ − p k − 1 I k − 1 = P ~ − 1 + 1 p k I k − 1 = [ 1 p 1 + 1 p k 0 0 ⋯ 0 0 1 p 2 + 1 p k 0 ⋯ 0 0 0 1 p 3 + 1 p k ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 ⋯ 1 p k − 1 + 1 p k ]
\begin{align*}
& \left( {{ 1 } \over { n }} \widetilde{\Sigma} \right)^{-1}
\\ =& \left( \widetilde{P} + \left( - \widetilde{\mathbf{p}} \widetilde{\mathbf{p}}^{T} \right) \right)^{-1}
\\ =& \widetilde{P}^{-1} + {{ \widetilde{P}^{-1} \widetilde{\mathbf{p}} \widetilde{\mathbf{p}}^{T} \widetilde{P}^{-1} } \over { 1 - \widetilde{\mathbf{p}}^{T} \widetilde{P}^{-1} \widetilde{\mathbf{p}} }}
\\ =& \widetilde{P}^{-1} + {{ I_{k-1} I_{k-1} } \over { 1 - I_{k-1} \widetilde{\mathbf{p}} }}
\\ =& \widetilde{P}^{-1} + {{ I_{k-1} I_{k-1} } \over { 1 - I_{k-1} \widetilde{\mathbf{p}} }}
\\ =& \widetilde{P}^{-1} + {{ 1 } \over { 1 - p_{1} - \cdots - p_{k-1} }} I_{k-1}
\\ =& \widetilde{P}^{-1} + {{ 1 } \over { p_{k} }} I_{k-1}
\\ =& \begin{bmatrix}
{{ 1 } \over { p_{1} }} + {{ 1 } \over { p_{k} }} & 0 & 0 & \cdots & 0
\\ 0 & {{ 1 } \over { p_{2} }} + {{ 1 } \over { p_{k} }} & 0 & \cdots & 0
\\ 0 & 0 & {{ 1 } \over { p_{3} }} + {{ 1 } \over { p_{k} }} & \cdots & 0
\\ \vdots & \vdots & \vdots & \ddots & \vdots
\\ 0 & 0 & 0 & \cdots & {{ 1 } \over { p_{k-1} }} + {{ 1 } \over { p_{k} }}
\end{bmatrix}
\end{align*}
= = = = = = = ( n 1 Σ ) − 1 ( P + ( − p p T ) ) − 1 P − 1 + 1 − p T P − 1 p P − 1 p p T P − 1 P − 1 + 1 − I k − 1 p I k − 1 I k − 1 P − 1 + 1 − I k − 1 p I k − 1 I k − 1 P − 1 + 1 − p 1 − ⋯ − p k − 1 1 I k − 1 P − 1 + p k 1 I k − 1 p 1 1 + p k 1 0 0 ⋮ 0 0 p 2 1 + p k 1 0 ⋮ 0 0 0 p 3 1 + p k 1 ⋮ 0 ⋯ ⋯ ⋯ ⋱ ⋯ 0 0 0 ⋮ p k − 1 1 + p k 1 S S S S = ∑ j = 1 k − 1 ( X j − n p j ) 2 n p j + ( ∑ j = 1 k − 1 ( X j − n p j ) ) 2 n p k = 1 n [ ∑ j = 1 k − 1 ( X j − n p j ) 1 p j ( X j − n p j ) + ∑ j = 1 k − 1 ( X j − n p j ) 1 p k ∑ j = 1 k − 1 ( X j − n p j ) ] = 1 n ( N ~ − n p ~ ) T ( 1 n Σ ~ ) − 1 ( N ~ − n p ~ ) = ( N ~ − n p ~ ) T ( Σ ~ − 1 ) ( N ~ − n p ~ )
\begin{align*}
S =& \sum_{j=1}^{k-1} {{ \left( X_{j} - n p_{j} \right)^{2} } \over { n p_{j} }} + {{ \left( \sum_{j=1}^{k-1} \left( X_{j} - n p_{j} \right) \right)^{2} } \over { n p_{k} }}
\\ =& {{ 1 } \over { n }} \left[ \sum_{j=1}^{k-1} \left( X_{j} - n p_{j} \right) {{ 1 } \over { p_{j} }} \left( X_{j} - n p_{j} \right) + \sum_{j=1}^{k-1} \left( X_{j} - n p_{j} \right) {{ 1 } \over { p_{k} }} \sum_{j=1}^{k-1} \left( X_{j} - n p_{j} \right) \right]
\\ =& {{ 1 } \over { n }} \left( \widetilde{N} - n \widetilde{\mathbf{p}} \right)^{T} \left( {{ 1 } \over { n }} \widetilde{\Sigma} \right)^{-1} \left( \widetilde{N} - n \widetilde{\mathbf{p}} \right)
\\ =& \left( \widetilde{N} - n \widetilde{\mathbf{p}} \right)^{T} \left( \widetilde{\Sigma}^{-1} \right) \left( \widetilde{N} - n \widetilde{\mathbf{p}} \right)
\end{align*}
S = = = = j = 1 ∑ k − 1 n p j ( X j − n p j ) 2 + n p k ( ∑ j = 1 k − 1 ( X j − n p j ) ) 2 n 1 [ j = 1 ∑ k − 1 ( X j − n p j ) p j 1 ( X j − n p j ) + j = 1 ∑ k − 1 ( X j − n p j ) p k 1 j = 1 ∑ k − 1 ( X j − n p j ) ] n 1 ( N − n p ) T ( n 1 Σ ) − 1 ( N − n p ) ( N − n p ) T ( Σ − 1 ) ( N − n p )
드 무아브르-라플라스 정리 : X i ∼ B ( 1 , p ) X_i \sim B(1,p) X i ∼ B ( 1 , p ) Y n = X 1 + X 2 + ⋯ + X n Y_n = X_1 + X_2 + \cdots + X_n Y n = X 1 + X 2 + ⋯ + X n Y n ∼ B ( n , p ) Y_n \sim B(n,p) Y n ∼ B ( n , p ) Y n − n p n p ( 1 − p ) → D N ( 0 , 1 )
{ { Y_n - np } \over {\sqrt{ np(1-p) } } }\overset{D}{\to} N(0,1)
n p ( 1 − p ) Y n − n p → D N ( 0 , 1 )
우리는 이미 Σ ~ \widetilde{\Sigma} Σ N ~ \widetilde{N} N ( N ~ − n p ~ ) ∼ ( 0 , Σ ~ )
\left( \widetilde{N} - n \widetilde{\mathbf{p}} \right) \sim \left( \mathbf{0} , \widetilde{\Sigma} \right)
( N − n p ) ∼ ( 0 , Σ ) 다변량정규분포 N k − 1 \mathcal{N}_{k-1} N k − 1 Σ ~ − 1 / 2 ( N ~ − n p ~ ) → D N k − 1 ( 0 , I k − 1 )
\widetilde{\Sigma}^{ - 1/2} \left( \widetilde{N} - n \widetilde{\mathbf{p}} \right) \overset{D}{\to} \mathcal{N}_{k-1} \left( \mathbf{0} , I_{k-1} \right)
Σ − 1/2 ( N − n p ) → D N k − 1 ( 0 , I k − 1 ) 표준정규분포 를 따르는 각 성분을 Z j ∼ N ( 0 , 1 ) Z_{j} \sim N \left( 0, 1 \right) Z j ∼ N ( 0 , 1 ) S S S 특성함수 를 ϕ \phi ϕ ( Z 1 , ⋯ , Z k − 1 ) \left( Z_{1} , \cdots ,Z_{k-1} \right) ( Z 1 , ⋯ , Z k − 1 ) I k − 1 I_{k-1} I k − 1 i ≠ j i \ne j i = j Z i Z_{i} Z i Z j Z_{j} Z j 독립 이다.
레비의 연속성 정리 : 가측공간 ( R d , B ( R d ) ) \left( \mathbb{R}^{d} , \mathcal{B} \left( \mathbb{R}^{d} \right) \right) ( R d , B ( R d ) ) n ∈ N ‾ n \in \overline{\mathbb{N}} n ∈ N 확률측도 를 μ n \mu_{n} μ n 특성함수 를 φ n \varphi_{n} φ n
(a): μ n \mu_{n} μ n μ ∞ \mu_{\infty} μ ∞ 약하게 수렴 한다. (b): 모든 t ∈ R d t \in \mathbb{R}^{d} t ∈ R d lim n → ∞ φ n ( t ) = φ ∞ ( t ) \lim_{n \to \infty} \varphi_{n} (t) = \varphi_{\infty} (t) n → ∞ lim φ n ( t ) = φ ∞ ( t ) ϕ \phi ϕ n → ∞ n \to \infty n → ∞ Z ∼ N ( 0 , 1 ) Z \sim N (0,1) Z ∼ N ( 0 , 1 ) ϕ ( t ) = E [ e i t S ] = E [ exp ( i t ( N ~ − n p ~ ) T Σ ~ − 1 ( N ~ − n p ~ ) ) ] → D E [ exp ( i t ( Z 1 2 + ⋯ + Z k − 1 2 ) ) ] = [ E [ exp ( i t Z 2 ) ] ] k − 1
\begin{align*}
\phi (t) =& E \left[ e^{itS} \right]
\\ =& E \left[ \exp \left( it \left( \widetilde{N} - n \widetilde{\mathbf{p}} \right)^{T} \widetilde{\Sigma}^{ - 1} \left( \widetilde{N} - n \widetilde{\mathbf{p}} \right) \right) \right]
\\ \overset{D}{\to} & E \left[ \exp \left( it \left( Z^{2}_{1} + \cdots + Z^{2}_{k-1} \right) \right) \right]
\\ = & \left[ E \left[ \exp \left( it Z^{2} \right) \right] \right]^{k-1}
\end{align*}
ϕ ( t ) = = → D = E [ e i tS ] E [ exp ( i t ( N − n p ) T Σ − 1 ( N − n p ) ) ] E [ exp ( i t ( Z 1 2 + ⋯ + Z k − 1 2 ) ) ] [ E [ exp ( i t Z 2 ) ] ] k − 1
카이제곱분포 의 성질:
표준정규분포의 제곱과의 관계 : X ∼ N ( μ , σ 2 ) X \sim N(\mu,\sigma ^2) X ∼ N ( μ , σ 2 ) V = ( X − μ σ ) 2 ∼ χ 2 ( 1 )
V=\left( { X - \mu \over \sigma} \right) ^2 \sim \chi ^2 (1)
V = ( σ X − μ ) 2 ∼ χ 2 ( 1 ) 적률생성함수 : m ( t ) = ( 1 − 2 t ) − r / 2 , t < 1 2 m(t) = (1-2t)^{-r/2} \qquad , t < {{ 1 } \over { 2 }} m ( t ) = ( 1 − 2 t ) − r /2 , t < 2 1 ϕ ( t ) → D [ E [ exp ( i t Z 2 ) ] ] k − 1 = [ 1 ( 1 − 2 i t ) 1 / 2 ] k − 1 = ( 1 − 2 i t ) − ( k − 1 ) / 2
\begin{align*}
\phi (t) \overset{D}{\to} & \left[ E \left[ \exp \left( it Z^{2} \right) \right] \right]^{k-1}
\\ =& \left[ {{ 1 } \over { \left( 1 - 2it \right)^{1/2} }} \right]^{k-1}
\\ =& (1-2it)^{-(k-1)/2}
\end{align*}
ϕ ( t ) → D = = [ E [ exp ( i t Z 2 ) ] ] k − 1 [ ( 1 − 2 i t ) 1/2 1 ] k − 1 ( 1 − 2 i t ) − ( k − 1 ) /2 S S S χ 2 ( k − 1 ) \chi^{2} \left( k-1 \right) χ 2 ( k − 1 )
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