Page 111 - Elements of Distribution Theory

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052184472Xc04 CUNY148/Severini May 24, 2005 2:39

4.2 Moments and Central Moments 97

Correlation
The correlation of X and Y, which we denote by ρ(X, Y), is defined by
Cov(X, Y)
ρ(X, Y) = ,
1
[Var(X)Var(Y)] 2
provided that Var(X) > 0 and Var(Y) > 0. The correlation is simply the covariance of
the random variables X and Y standardized to have unit variance. Hence, if a, b, c, d are
constants, then the correlation of aX + b and cY + d is the same as the correlation of X
and Y.
The covariance and correlation are measures of the strength of the linear relationship
between X and Y; the correlation has the advantage of being easier to interpret, because of
the following result.

2
Theorem 4.3. Let X and Y denote real-valued random variables satisfying E(X ) < ∞
2
and E(Y ) < ∞. Assume that Var(X) > 0 and Var(Y) > 0. Then
2
(i) ρ(X, Y) ≤ 1
2
(ii) ρ(X, Y) = 1 if and only if there exist real-valued constants a, b such that
Pr(Y = aX + b) = 1.
(iii) ρ(X, Y) = 0 if and only if, for any real-valued constants a, b,
2
E{[Y − (aX + b)] }≥ Var(Y).

Proof. Let Z = (X, Y), g 1 (Z) = X − µ X , and g 2 (Z) = Y − µ Y , where µ X = E(X) and
µ Y = E(Y). Then, by the Cauchy-Schwarz inequality,
2
2
2
E[g 1 (Z)g 2 (Z)] ≤ E[g 1 (Z) ]E[g 2 (Z) ].
That is,
2
E[(X − µ X )(Y − µ Y )] ≤ Var(X)Var(Y);
part (i) follows.
2
The condition ρ(X, Y) = 1is equivalent to equality in the Cauchy-Schwarz inequality;
under the conditions of the theorem, this occurs if and only there exists a constant c such
that
Pr(Y − µ Y = c(X − µ X )) = 1.
This proves part (ii).
By part (iv) of Theorem 4.1, for any constants a and b,
2 2
E{[Y − (aX + b)] }≥ E{[Y − µ Y − a(X − µ X )] }
.
2
= Var(Y) + a Var(X) − 2a Cov(Y, X)
If ρ(X, Y) = 0, then Cov(Y, X) = 0so that, by Theorem 4.1,
2 2
E{[Y − (aX + b)] }≥ Var(Y) + a Var(X) ≥ Var(Y).
Now suppose that, for any constants a, b,
2
E{[Y − (aX + b)] }≥ Var(Y).