Question

A circle of radius r has area A = πr . If a random circle has a radius that is uniformly distributed on the interval (0, 2) what is the mean of the area of the circle?

Answer 1

solution O circle of radius & has area A= Tr? We need to catu calculate E(A) XNucaib) fix acxcb =1 L (b-a) to sowo In our case interval (a,b) = (0,2) Hence, for) = 2 2 10 = 2 Fr= L 2 -0 oarea 0<r<2 ; elsewhere

let CO - Elr) = Sr. F(0) dx - Since Fr)=0 for Fre (-0,o) U (2,00) Hencer E(A)= E[T2) EMTE [72) = π 2 fio dr Alt f(x) = 1 / 2 vre(0,2) since ETR3). E(8²) = / 4 (22) dr = Y3 [1/2]?

= { El82) = Hences bar) 7 - ( 5) C0.00 ECA= TEG2) = π(4/3) = 4IT 3 stje T ELA) = 45 3