9 an ecologist wishes to mark off a circular sampling region

9) an ecologist wishes to mark off a circular sampling region having radious 10 meters. Howerever, th radius of the resulting region is actully a random variable R with PDF f(r)=0.5 0< =  r <= 11, f(r)=0 otherwise. what is the expected area of the resulting region? Note: In formulating your answer, take into account that it is generally the cae that for a random variable X and function g: Eg(x) not equal to g(EX).

Solution

Solution : 30.25

Expected area of resulting region :

E(X) = integral (from 0 to 11) r * f(r) dr

       = integral (from 0 to 11) ( 0.5 r )dr

       = (from 0 to 11) 0.5 * (r^2 /2)

       = 0.25 ( 121 - 0)

       = 30.25 Answer