EXAMPLE 5.14

In EXAMPLE 5.5, the joint pdf of the amount $X$ of almonds and amount $Y$ of cashews in a 1-lb can of nuts was

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

If 1 lb of almonds costs the company \ {1.50}$,$1\mathrm{{lb}}$ofcashewscosts$$ {2.25}$,and$1\mathrm{{lb}}$ofpeanutscosts$$ {.75}$, then the total cost of the contents of a can is

$$h\left(X,Y\right)=\left(1.5\right)X+\left(2.25\right)Y+\left(.75\right)\left(1-X-Y\right)=.75+.75X+1.5Y$$

(since $1-X-Y$ of the weight consists of peanuts). The expected total cost is

$$\begin{array}{rl}E[h(X,Y)]& ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }h(x,y)\cdot f(x,y)dxdy\\ & ={\int }_0^1{\int }_0^{1-x}\left(0.75+0.75x+1.5y\right)\cdot 24xydydx=\$1.65\end{array}$$