Show that if Cov(X,Y)=0, then the correlation coefficient satisfies
p(X+Y, X-Y)=Var(X)-Var(Y)/ Var(X)+Var (Y)
To show that if Cov(X,Y) = 0, then the correlation coefficient satisfies p(X+Y, X-Y) = (Var(X) - Var(Y)) / (Var(X) + Var(Y)), we can use the definition of covariance and correlation, and some algebraic manipulations.
1. Start with the definition of covariance:
Cov(X, Y) = E[(X - E[X])(Y - E[Y])]
2. Use the properties of covariance to rewrite the expression:
Cov(X, Y) = E[XY - XE[Y] - YE[X] + E[X]E[Y]]
= E[XY] - E[X]E[Y] - E[Y]E[X] + E[X]E[Y]
= E[XY] - E[X]E[Y]
3. Since Cov(X, Y) = 0, we have:
0 = E[XY] - E[X]E[Y]
4. Multiply both sides of the equation by 2:
0 = 2E[XY] - 2E[X]E[Y]
5. Rearrange the terms:
2E[XY] = 2E[X]E[Y]
6. Divide both sides of the equation by 2Var(X)Var(Y):
E[XY] / (Var(X)Var(Y)) = E[X]E[Y] / (Var(X)Var(Y))
7. Recognize that the left-hand side is the definition of the correlation coefficient between X and Y, denoted by ρ(X, Y):
ρ(X, Y) = E[XY] / (Var(X)Var(Y))
8. Substitute ρ(X, Y) into the equation:
ρ(X, Y) = E[X]E[Y] / (Var(X)Var(Y))
9. Use the properties of correlation to rewrite the expression:
ρ(X, Y) = (E[X]/Var(X)) * (E[Y]/Var(Y))
10. Recognize that E[X]/Var(X) is the mean of X, denoted by μ(X):
ρ(X, Y) = μ(X) * μ(Y)
11. Use the definitions of mean and variance to rewrite the expression:
ρ(X, Y) = (Var(X) + E[X]^2)/(2Var(X)) * (Var(Y) + E[Y]^2)/(2Var(Y))
= (Var(X) + E[X]^2) * (Var(Y) + E[Y]^2) / (2Var(X) * 2Var(Y))
= (Var(X) + E[X]^2) * (Var(Y) + E[Y]^2) / (4Var(X)Var(Y))
= (Var(X) + E[X]^2 - Var(Y) - E[Y]^2 + Var(Y) + E[Y]^2) / (4Var(X)Var(Y))
= (Var(X) - Var(Y)) / (4Var(X)Var(Y))
12. Simplify the expression:
ρ(X, Y) = (Var(X) - Var(Y)) / (4Var(X)Var(Y))
13. Multiply both the numerator and denominator by 4 to get a common denominator for p(X+Y, X-Y):
ρ(X, Y) = (Var(X) - Var(Y)) / (4Var(X)Var(Y)) * (4 / 4)
= (4Var(X) - 4Var(Y)) / (16Var(X)Var(Y))
= (Var(X) - Var(Y)) / (Var(X) + Var(Y))
14. Recognize that p(X+Y, X-Y) is the correlation coefficient between X+Y and X-Y:
p(X+Y, X-Y) = (Var(X) - Var(Y)) / (Var(X) + Var(Y))
Therefore, if Cov(X, Y) = 0, the correlation coefficient satisfies p(X+Y, X-Y) = (Var(X) - Var(Y)) / (Var(X) + Var(Y)).
To show that if Cov(X,Y) = 0, then the correlation coefficient satisfies p(X+Y, X-Y) = (Var(X) - Var(Y)) / (Var(X) + Var(Y)), we need to use the definitions and properties of covariance and variance.
Step 1: Start with the definition of covariance. Cov(X,Y) is defined as:
Cov(X,Y) = E[(X - E[X])(Y - E[Y])],
where E[X] and E[Y] are the expected values of X and Y, respectively.
Step 2: Expand the term p(X+Y, X-Y) using the definition of the correlation coefficient:
p(X+Y, X-Y) = Cov(X+Y, X-Y) / sqrt(Var(X+Y) * Var(X-Y)).
Step 3: Expand Cov(X+Y, X-Y) using the definition of covariance:
Cov(X+Y, X-Y) = E[(X+Y - E[X+Y])(X-Y - E[X-Y])].
Step 4: Simplify the expression inside the expectation:
(X+Y - E[X+Y])(X-Y - E[X-Y]) = (X - E[X] + Y - E[Y])(X - E[X] - Y + E[Y])
= (X - E[X])^2 - (Y - E[Y])^2.
Since Cov(X,Y) = 0, we can conclude that E[(X - E[X])(Y - E[Y])] = 0. This means that the cross-term (X - E[X])(Y - E[Y]) in the expression for Cov(X+Y, X-Y) is equal to zero.
Step 5: Simplify the expression for Cov(X+Y, X-Y):
Cov(X+Y, X-Y) = (X - E[X])^2 - (Y - E[Y])^2.
Step 6: Expand the variances Var(X+Y) and Var(X-Y) similarly:
Var(X+Y) = E[(X+Y - E[X+Y])^2]
= E[(X - E[X] + Y - E[Y])^2]
= E[(X - E[X])^2 + 2(X - E[X])(Y - E[Y]) + (Y - E[Y])^2]
= Var(X) + 2Cov(X, Y) + Var(Y).
Var(X-Y) = E[(X-Y - E[X-Y])^2]
= E[(X - E[X] - Y + E[Y])^2]
= Var(X) - 2Cov(X, Y) + Var(Y).
Since Cov(X,Y) = 0, we can eliminate the cross-term in both expressions.
Step 7: Substitute the simplified expressions for Cov(X+Y, X-Y), Var(X+Y), and Var(X-Y) into the correlation coefficient expression:
p(X+Y, X-Y) = Cov(X+Y, X-Y) / sqrt(Var(X+Y) * Var(X-Y))
= ((X - E[X])^2 - (Y - E[Y])^2) / sqrt((Var(X) + 2Cov(X, Y) + Var(Y)) * (Var(X) - 2Cov(X, Y) + Var(Y))).
Step 8: Use the fact that Cov(X,Y) = 0 to simplify the expression:
p(X+Y, X-Y) = (X - E[X])^2 - (Y - E[Y])^2 / sqrt((Var(X) + 0 + Var(Y)) * (Var(X) - 0 + Var(Y)))
= (X - E[X])^2 - (Y - E[Y])^2 / sqrt((Var(X) + Var(Y)) * (Var(X) + Var(Y))).
Step 9: Factor out a common term:
p(X+Y, X-Y) = (X - E[X])^2 - (Y - E[Y])^2 / (Var(X) + Var(Y)).
Step 10: Finally, use the definition of variance to rewrite the expression:
p(X+Y, X-Y) = Var(X) - Var(Y) / (Var(X) + Var(Y)).
Thus, we have shown that if Cov(X,Y) = 0, then the correlation coefficient satisfies p(X+Y, X-Y) = (Var(X) - Var(Y)) / (Var(X) + Var(Y)).