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?

Question

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

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Answer 1 · 2018-06-06T11:20:34

Option: C

#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:

Expected value of #r#:

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

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