# 11.4. Bounds on Correlation#

For a random pair $$(X, Y)$$, the correlation $$r(X, Y)$$ is defined as

$r(X, Y) ~ = ~ E\Bigg( \left(\frac{X-\mu_X}{\sigma_X}\right)\left(\frac{Y-\mu_Y}{\sigma_Y}\right) \Bigg) ~ = ~ E(X^*Y^*)$

where $$X^*$$ and $$Y^*$$ are respectively $$X$$ and $$Y$$ measured in standard units.

Thus by definition, correlation is a pure number and has no units.

You have seen several properties of correlation in Data 8. Some are obvious, such as $$r(X, Y) = r(Y, X)$$. Some require proof.

In this brief section we will prove one principal property, which is that correlation is a number between $$-1$$ and $$1$$. You will prove a few other properties in exercises. In the next section we will specify the sense in which correlation measures clustering about a straight line.

## 11.4.1. Lower Bound#

As a preliminary, recall that

$E(X^*) = 0, ~~~ Var(X^*) = 1 = E\left({X^*}^2\right)$

So also $$E(Y^*) = 0$$ and $$E\left({Y^*}^2\right) = 1$$.

We know the expected squares, and what we need is a bound on the expected product $$E(X^*Y^*)$$. A result that connects the squares and the product of two numbers is $$(a+b)^2 = a^2 + 2ab + b^2$$.

So let’s find $$E\left((X^* + Y^*)^2\right)$$ and see what that gives us.

\begin{split} \begin{align*} E\left((X^* + Y^*)^2\right) ~ &= ~ E\left({X^*}^2\right) + 2E(X^*Y^*) + E\left({Y^*}^2\right) \\ &= ~ 2 + 2E(X^*Y^*) \end{align*} \end{split}

Since $$E\big{(}(X^* + Y^*)^2\big{)} \ge 0$$, we have

$2 + 2E(X^*Y^*) \ge 0$

which is the same as

$E(X^*Y^*) \ge -1$

## 11.4.2. Upper Bound#

Play the same game with $$E\big{(}(X^* - Y^*)^2\big{)}$$ to see that

$2 - 2E(X^*Y^*) \ge 0$

which is the same as

$E(X^*Y^*) \le 1$

because division by $$-2$$ flips the direction of the inequality.

## 11.4.3. Other Properties#

As you know from Data 8, correlation measures linear association. In exercises you will show that if $$Y$$ is a linear function of $$X$$ then $$r(X, Y)$$ is either $$1$$ or $$-1$$.

You will also find the relation between $$r(X, Y)$$ and $$r(X, W)$$ where $$W$$ is a linear function of $$Y$$.

In the next section we will return to regression and formalize the idea that correlation measures clustering about a straight line. Our result will imply that if $$r(X, Y)$$ is either $$1$$ or $$-1$$, then the relation between $$X$$ and $$Y$$ must be perfectly linear.