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An ecologist wishes to mark off a circular sampling region having a radius $10{m}$. However, the radius of the resulting region is actually a random variable $R$ with pdf

$f(r)=\left\{\begin{array}{cl} \frac{3}{4}\left[1-(10-r)^2\right] & 9 \leq r \leq 11 \\ 0 & \text { otherwise } \end{array}\right.$

What is the expected area of the resulting circular region?

Solution

VerifiedThe radius of the region is a random variable $R$ with pdf

$f(r)=\begin{cases} \cfrac{3}{4}\left[1-(10-r)^2 \right] &; 9\le r\le 11\\ 0&;\text{otherwise} \end{cases}$

The area of the region is $A(R)=\pi R^2$ We need to find the expected area of the region i.e.,

$\mathbb{E}(A(R))=\int_{-\infty}^{\infty}{A(r)f(r)dr}$

The calculations are as follows.

$\begin{align*} \mathbb{E}(A(R))&=\int_{-\infty}^{\infty}{A(r)f(r)dr}\\ &=\int_{9}^{11}{\cfrac{3\pi}{4} r^2\left[1-(10-r)^2 \right]dr}\\ &=\cfrac{3\pi}{4} \left. \left[-\cfrac{r^5-25r^4+165r^3}{5} \right] \right|_9^{11}\\ &=\cfrac{3\pi}{4}\left[-\cfrac{14641}{5} +\cfrac{15309}{5} \right]\\ &\approx 314.7876 \end{align*}$

Hence, the expected area of the region is $314.7876$.

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