What is the expected area of this enclosure?

Question

I have no clue how to do this textbook question: An environmental artist is planning to construct a rectangle with 36m of fencing as part of an outdoor installation. If the length of the rectangle is a randomly chosen integral number of $L$ metres, what is the expected area of this enclosure?

Answer 1

Let $L$ denote the length of the rectangle, and $W$ the width.

We know that $2L+2W=36$, so $L+W=18$, i.e. $W=18-L$. The rectangle's area is $A=LW=L(18-L)$.

Since $L$ is known to be an integer, and $W,L$ must both be nonnegative, then $L$ is chosen from $$\{0,1,2,3,\ldots, 18\}$$

Now, computed the expected value of $A$, as $L$ varies among its possible values. Do you need help with this part?

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Adding more details per request. The desired value is $$\sum_{L}APr(A)=\sum_L L(18-L)\frac{1}{19}=\frac{1}{19}\sum_{L=0}^{18} 18L - L^2=$$ $$=\frac{18}{19}\sum_{L=0}^{18}L - \frac{1}{19}\sum_{L=0}^{18}L^2$$

These last sums can be computed using Faulhaber's formulas.