Question

An ecologist wishes to mark off a circular sampling region having radius 5 m. However, the radius of the resulting region is actually a random variable R with the following pdf.

A={}{:- 15-12] otherwise

What is the expected area of the resulting circular region? (Round your answer to two decimal places.)

Answer 1

Given the radius of the resulting region is actually a random variable R with the following pdf.

A={}{:- 15-12] otherwise

Expected area of the resulting circular region:

$\int rf(r)dr$

$\Rightarrow \int_{4}^{6} r\frac{3}{4}[1-(5-r)^2]dr$

$\Rightarrow \frac{3}{4}\int_{4}^{6} r[1-(5-r)^2]dr$

$\Rightarrow \frac{3}{4}\int_{4}^{6} r[1-25-r^2+10r]dr$

$\Rightarrow \frac{3}{4}\int_{4}^{6} [-24r-r^3+10r^2]dr$

$\Rightarrow \frac{3}{4}[-24\frac{r^2}{2}-\frac{r^4}{4}+10\frac{r^3}{3}]_{4}^{6}$

$\Rightarrow 5$