91Ó°ÊÓ

Probability particular outcomes—how likely they are—when we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

Assume that

\(\text{Y=aX+b,}\text{a }\!\!{}^\text{1}\!\!\text{0}\text{.}\)

The

correlation coefficient

of \({\rm{X}}\) and \({\rm{Y}}\)is

\({{\rm{\rho }}_{{\rm{X,Y}}}}{\rm{ = Corr(X,Y) = }}\frac{{{\rm{Cov(X,Y)}}}}{{{{\rm{\sigma }}_{\rm{X}}}{\rm{ \times }}{{\rm{\sigma }}_{\rm{Y}}}}}\)

The following holds

\(\begin{aligned}{\rm{Corr(X,aX + b) = }}\frac{{{\rm{Cov(X,aX + b)}}}}{{\sqrt {{\rm{\sigma }}_{\rm{X}}^{\rm{2}}{\rm{ \times \sigma }}_{{\rm{aX + b}}}^{\rm{2}}} }}\\\mathop {\rm{ = }}\limits^{{\rm{(1)}}} \frac{{{\rm{a\sigma }}_{\rm{X}}^{\rm{2}}}}{{\sqrt {{\rm{\sigma }}_{\rm{X}}^{\rm{2}}{\rm{ \times }}{{\rm{a}}^{\rm{2}}}{\rm{\sigma }}_{\rm{X}}^{\rm{2}}} }}\\{\rm{ = }}\frac{{{\rm{a\sigma }}_{\rm{X}}^{\rm{2}}}}{{{\rm{a}}\mid {\rm{\sigma }}_{\rm{X}}^{\rm{2}}}}\\\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\rm{ \pm 1,}}\end{aligned}\)

(1): here we use

\(\begin{aligned}{\rm{Cov(X,Y) = Cov(X,aX + b)}}\\{\rm{ = E(X(aX + b)) - E(X) \times E(aX + b)}}\\{\rm{ = a}}\left( {{\rm{E}}\left( {{{\rm{X}}^{\rm{2}}}} \right){\rm{ - (E(X)}}{{\rm{)}}^{\rm{2}}}} \right)\\{\rm{ = aV(X)}}\\{\rm{ = a\sigma }}_{\rm{X}}^{\rm{2}}\end{aligned}\)

and rules of variance,

(2): if \({\rm{a > 0}}\)then

\({\rm{Corr(X,aX + b) = 1}}\)

if\({\rm{a < 0}}\), then

\({\rm{Corr(X,aX + b) = - 1}}\)