Distance to the centre

Topic: Random variables and distributions · more questions on this topic
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A point is drawn uniformly from the disc of radius 4. Let R be its distance to the centre.

1. Find \mathbb{P}(R>2).

2. Find the density of R.

Worked answer (click to show)

1. The disc has area 16\pi and the disc of radius 2 has area 4\pi, so \mathbb{P}(R>2)=(16\pi-4\pi)/(16\pi)=3/4.

2. For 0<r<4, \mathbb{P}(R\le r)=\pi r^2/(16\pi)=r^2/16. Differentiating gives f(r)=r/8 on (0,4), and f(r)=0 elsewhere.