Independent solution

How to solve this Joint Distributions question

Setup

Setup

Let R denote the daily revenue. The supplied probability rule produces mass numerators 2, 4, 6 in the first row and 4, 2, 0 in the second row, all over 18.

R=XYR=XY
18pX,Y(x,y)=1012141246242018p_{X,Y}(x,y)=\begin{array}{c|ccc} &10&12&14\\ \hline 1&2&4&6\\ 2&4&2&0\end{array}

Model

Model

For a function of two discrete variables, multiply its value at each support point by the joint mass at that point.

E⁡[R]=E⁡[XY]=∑x∑yxy pX,Y(x,y)\operatorname{E}[R]=\operatorname{E}[XY]=\sum_x\sum_y xy\,p_{X,Y}(x,y)

Compute

Compute

Evaluate the six weighted products and combine their numerators.

E⁡[XY]=20+48+84+80+48+018\operatorname{E}[XY]=\frac{20+48+84+80+48+0}{18}
E⁡[XY]=28018=1409=15.555…\operatorname{E}[XY]=\frac{280}{18}=\frac{140}{9}=15.555\ldots

Answer

Answer

The expected revenue is 140/9.

1409(A)\boxed{\frac{140}{9}\quad\text{(A)}}