EXAMPLE 5.7 (Example 5.5 continued)

EXAMPLE 5.5 continued.

Because $f\left(x,y\right)$ has the form of a product, $X$ and $Y$ would appear to be independent. However, although $f_X\left(3/4\right)=f_Y\left(3/4\right)=9/16$, $f\left(3/4,3/4\right)=0\ne 9/16\cdot 9/16$, so the variables are not in fact independent. To be independent, $f\left(x,y\right)$ must have the form $g\left(x\right)\cdot h\left(y\right)$ and the region of positive density must be a rectangle whose sides are parallel to the coordinate axes.