EXAMPLE 5.16 (Example 5.5 were continued)

EXAMPLE 5.5 were continued

The joint and marginal pdf’s of $X=$ amount of almonds and $Y=$ amount of cashews

$$f\left(x,y\right)=\{\begin{array}{cc}24xy& 0\le x\le 1,0\le y\le 1,x+y\le 1\\ 0& \text{ otherwise }\end{array}$$ $$f_X\left(x\right)=\{\begin{array}{cc}12x{\left(1-x\right)}^2& 0\le x\le 1\\ 0& \text{ otherwise }\end{array}$$

with $f_Y\left(y\right)$ obtained by replacing $x$ by $y$ in $f_X\left(x\right)$. It is easily verified that ${\mu }_X={\mu }_Y=\frac{2}{5}$, and

$$\begin{array}{rl}E\left(XY\right)& ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }xyf\left(x,y\right)dxdy\\ & ={\int }_0^1{\int }_0^{1-x}xy\cdot 24xydydx\\ & =8{\int }_0^1{x}^2{\left(1-x\right)}^{3}dx\\ & =\frac{2}{15}.\end{array}$$

Thus

$$\begin{array}{rl}\mathrm{Cov}(X,Y)& =\frac{2}{15}-\left(\frac{2}{5}\right)\left(\frac{2}{5}\right)\\ & =\frac{2}{15}-\frac{4}{25}\\ & =-\frac{2}{75}.\end{array}$$

A negative covariance is reasonable here because more almonds in the can implies fewer cashews.