Question

5. * Suppose that a point is chosen at random in the interior of a circle of radius 1. Let D be the distance of the selected point from the centre of the circle. What is the density function of D?

Answer 1

Let X and Y be the coordinates of the randomly chosen point. We assume that the point is uniformly distmibuted within the circle of radius 1. of X The joint pdf and Y is 2 a ty f(x,y) = ses O 0.W. As funny) is the joint Y, so Х and pdf of have we ss finy) andy al

- ov, sse e dndy x² + y² = 1 - OV, ss dudy x² + y² = 1 or, Issandy are a of The the region A, where x² + y²1 A = {(x, y): x² + y² =1} of E the area .: The region the circle - The area of radius 1 with 2 nx (1)

Ssandy = 1. u? y2=1 - J ov, af of x .: The int joi Y is and x² + y ² cl - fin, y ) = o o.w.

assume we that the circle is Here center of the at origin. 2 x² + y So, D of ID is The ploed) Fold) Р UX x² + y? y? d pl x² + y² e d²)

dco ss fixrylandy x² + y²ed² I oldal Now, ss finy) andy x² + y² <d² ss ħ andy 2 2 2 andy a SS C 2 x² + y² = ď 2 2 M d

[ SS dudy The area of 2. circle with 2 n d radius r. d² . il 0 da L d) 2 d? Ioada d > 1

of D is pdf The d F (n) f (n) d. n d (N²), o anal du 0 0.W. 2 n O L n cl 0 O.W.