91Ó°ÊÓ

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of 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.

(a):

Proposition:

The following holds

\({\rm{Cov(X,Y) = E(XY) - E(X) \times E(Y)}}\)

Using this, the following holds

\(\begin{aligned}{\rm{Cov(aX + b,cY + d) = E((aX + b) \times (cY + d)) - E(aX + b) \times E(cY + d)}}\\{\rm{ = E(acXY + adX + bcY + bd) - (aE(X) + b) \times (cE(Y) + d)}}\\{\rm{ = acE(XY) + adE(X) + bcE(Y) + bd - (acE(X)E(Y) + adE(X) + bcE(Y) + bd)}}\\{\rm{ = acE(XY) - acE(X)E(Y)}}\\{\rm{ = ac(E(XY) - E(X)E(Y))}}\\{\rm{ = acCov(X,Y)}}\end{aligned}\)

(b):

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

\(\begin{aligned}{\rm{Corr(aX + b,cY + d) = }}\frac{{{\rm{Cov(aX + b,cY + d)}}}}{{{{\rm{\sigma }}_{{\rm{aX + b}}}}{\rm{ \times }}{{\rm{\sigma }}_{{\rm{cY + d}}}}}}\\\mathop {\rm{ = }}\limits^{{\rm{(1)}}} \frac{{{\rm{acCov(X,Y)}}}}{{{\rm{|a|}}{{\rm{\sigma }}_{\rm{X}}}{\rm{ \times |c|}}{{\rm{\sigma }}_{\rm{Y}}}}}\\{\rm{ = }}\frac{{{\rm{ac}}}}{{{\rm{|ac|}}}}{\rm{Corr(X,Y),}}\end{aligned}\)

(1): here we used \({\rm{(a)}}\) and the properties (rules) of the standard deviation.

Notice that when \({\rm{a}}\) and \({\rm{c}}\) have the same sign we have that\({\rm{ac = |ac|}}\), and

\({\rm{Corr(aX + b,cY + d) = }}\frac{{{\rm{ac}}}}{{{\rm{ac}}}}{\rm{Corr(X,Y) = Corr(X,Y)}}{\rm{. }}\)

(c):

If \({\rm{a}}\) and \({\rm{c}}\) have opposite signs, we have that\({\rm{|ac| = - ac}}\), from which we get

\({\rm{Corr(aX + b,cY + d) = }}\frac{{{\rm{ac}}}}{{{\rm{ - ac}}}}{\rm{Corr(X,Y) = - Corr(X,Y)}}\)

Therefore, the solution is \({\rm{Corr(aX + b,cY + d) = - Corr(X,Y)}}\).