If-you-pick-two-random-points-on-a-circle-s-circumference-with-radius-1-what-is-the-average-area-between-the-two-points-and-the-origin-

Question Number 6672 by FilupSmith last updated on 11/Jul/16

If you pick two random points on a {circle}^{'} s

circumference with radius 1, what is

the average area between the

two points and the origin ?

Commented by Yozzii last updated on 11/Jul/16

A v e r a g e a r e a, A (r) = \frac{1}{π - 0} \int_{0}^{π} \frac{1}{2} r^{2} s i n θ d θ

A (r) = \frac{- r^{2}}{2 π} c o s θ ∣_{0}^{π} = \frac{r^{2}}{π} .

r = 1 \Rightarrow A (1) = \frac{1}{π}

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E (A) = E (\frac{1}{2} r^{2} s i n θ) = \int_{0}^{π} \frac{1}{2} r^{2} s i n θ f (θ) d θ

p r o b a b i l i t y d e n s i t y f u n c t i o n f (θ) = {\begin{cases} 1 / π 0 ⩽ θ ⩽ π \\ 0 o t h e r w i s e \end{cases}

E (A) = \frac{r^{2}}{2 π} \int_{0}^{π} s i n θ d θ = \frac{r^{2}}{π}

Commented by FilupSmith last updated on 11/Jul/16

why is the integral between 0 and π

rather than 0 and 2 π ?

Commented by Yozzii last updated on 11/Jul/16

I l e t t h e o r i g i n b e t h e c e n t r e o f t h e

c i r c l e s o t h a t 0 ⩽ ∠ P_{1} {O P}_{2} ⩽ π a s i s

e x p e c t e d o f a t r i a n g l e .

Commented by FilupSmith last updated on 11/Jul/16

I u n d e r s t a n d! T h a n k s