EXAMPLE 5.18

Let $X$ and $Y$ be discrete rv’s with joint pmf

$$p\left(x,y\right)=\{\begin{array}{ll}.25& \left(x,y\right)=\left(-4,1\right),\left(4,-1\right),\left(2,2\right),\left(-2,-2\right)\\ 0& \text{ otherwise }\end{array}$$

The points that receive positive probability mass are identified on the $\left(x,y\right)$ coordinate system in Figure 5.5.

Figure 5.5 The population of pairs for Example 5.18

It is evident from the figure that the value of $X$ is completely determined by the value of $Y$ and vice versa, so the two variables are completely dependent. However, by symmetry ${\mu }_X={\mu }_Y=0$ and $E\left(XY\right)=\left(-4\right)\left(.25\right)+\left(-4\right)\left(.25\right)+\left(4\right)\left(.25\right)+\left(4\right)\left(.25\right)=0.$ The covariance is then $\mathrm{Cov}\left(X,Y\right)=E\left(XY\right)-{\mu }_X\cdot {\mu }_Y=0$ and thus ${\rho }_{X,Y}=0.$ Although there is perfect dependence, there is also complete absence of any linear relationship! A value of $\rho$ near 1 does not necessarily imply that increasing the value of $X$ causes $Y$ to increase. It implies only that large $X$ values are associated with large $Y$ values.

For example, in the population of children,

• vocabulary size and number of cavities are quite positively correlated,
  • but it is certainly not true that cavities cause vocabulary to grow.
• Instead, the values of both these variables tend to increase as the value of age, a third variable, increases.
• For children of a fixed age, there is probably a low correlation between number of cavities and vocabulary size.
• In summary, association (a high correlation) is not the same as causation.