A stone is thrown into a circular pond of radius 1 meter. Suppose the stone falls uniformly at random on the area of the pond. What will be the expected distance of the stone from the center of the pond?

Options are:
A. 1/3
B. 1/2
C. 2/3
D. 1/2^(1/2) 1212

1 Answer
Jun 6, 2018

Option: C

Explanation:

A(r) = pi r^2, qquad dA(r) = 2 pi r \ dr

By definition of the PDF mathbb P(r):

mathbb P(r) * dr = P(r lt X lt r + dr)

= (dA)/(pi * 1^2) = 2 r dr implies mathbb P(r) = 2 r

Check:

  • int_0^1 mathbb P \ dr = 1

Expected value of r:

langle r rangle = int_0^1 \ r \ mathbb P \ dr

= 2 int_0^1 r^2 \ dr = 2/3