統計學─相關係數證明題?

由以下推導過程可知, 題目有誤,
若 b、d < 0 應更正為 bd < 0
若 b、d > 0 應更正為 bd > 0

pf :
Suu
= Σ U² - (1/n)( Σ U )²
= Σ (a+bx)² - (1/n)*[ Σ (a+bx) ]²
= Σ ( a² + 2abx + b²x² ) - (1/n)( an + bΣx )²
= a²n + 2abΣx + b²Σx² - (1/n)[ a²n² + 2abnΣx + b²( Σx )² ]
= a²n + 2abΣx + b²Σx² - a²n - 2abΣx - (1/n)b²( Σx )²
= b² * [ Σx² - (1/n)( Σx )² ]
= b² * Sxx

同理可得 Sv v = d² * Syy

Suv
= Σ UV - (1/n)( ΣU )( ΣV )
= Σ (a+bx)(c+dy) - (1/n)*[ Σ (a+bx) ]*[ Σ (c+dy) ]
= Σ ( ac + ady + bcx + bdxy ) - (1/n)*( an + bΣx )( cn + dΣy )
= Σ ( ac + ady + bcx + bdxy ) - (1/n)*( acn² + adnΣy + bcnΣx + bdΣxΣy )
= acn + adΣy + bcΣx + bd( Σ xy ) - acn - adΣy - bcΣx - (1/n)bdΣxΣy
= bd( Σ xy ) - (1/n)bdΣxΣy
= bd * [ Σ xy - (1/n)ΣxΣy ]
= bd * Sxy

r uv
= Suv / √( Suu * Sv v )
= bd * Sxy / √( b² * Sxx * d² * Syy )
= ( bd / ｜bd｜) * Sxy / √( Sxx * Syy )
= ( bd / ｜bd｜) * r xy

因此 :
r uv = ( bd / ｜bd｜) * r xy

當 bd < 0
r uv = ( bd / ｜bd｜) * r xy = [ bd / ( - bd ) ] * r xy = - r xy

當 bd > 0
r uv = ( bd / ｜bd｜) * r xy = ( bd / bd ) * r xy = r xy

Q.E.D.