Consider a circle of radius 1 with center at the point A = (1; 0). Let O denote the origin. A point B is chosen at random on the circle (not inside). Let X denote the X-coordinate of the point B. Find the pdf of X.

Consider the points on the circle in polar coordinates: all points on the circle are a fixed distance away from the center, and their coordinates are determined entirely by an angle made relative to the positive -axis.

Any point B on the circle has -coordinate . Let be a random variable uniformly distributed on the interval . Then

Now,

so that the density of is