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

step-by-step explanation:

this can be solved first with pythagorean theorem, which tells us to square both legs of the triangle and sum it. this will be equal to the square of the hypotenuse.

the hypotenuse is the side "opposite" the 90 degree angle, which is x.

so we can write using pythagorean theorem,

simplifying:

to further simplify this radical, we need to use the property  

thus