Question

A dart is randomly thrown at a dartboard of diameter 24. (By randomly we mean that regions of equal area on the dartboard are equally likely to be struck by the dart).

Let D be the distance from the center of the dartboard from the point where the dart strikes. Find E[D].

Answer 1

We are given here that regions of equal area on the dartboard are equally likely to be struck by the dart.

The cumulative distribution function for D here is obtained as:

P(D <= d) = Area proportion of circle with radius D of the total area

Therefore the PDF for d is obtained by differentiating the CDF with respect to d as:

Therefore the expected value of D here is computed as:

Therefore 8 is the required expected value of D here