Why is the distribution of the x coordinate of a point randomly selected from the circumference the unit cricle not uniform?

Question

I am currently attempting to solve a question which deals with the distribution of the $x$ and $y$ coordinates of a randomly chosen point from the circumference of the unit circle. When I first attempted the question I thought that since point is chosen at random, it would follow that it is uniformly distributed on the circumference of the circle, and moreover, its $x$ and $y$ coordinates will be uniformly distributed on the region $[-1,1]$. However, this is wrong.

I have found a couple of threads in regards to this issue, where the answers suggest one uses the fact that $x=\cos \theta$ and start from there. My issue is that this was not my original thought and that without going through the algebra with $x=\cos \theta$ I wouldn't be able to say why the distribution of $x$ (and $y$ for that mater) are not uniform.

Could someone explain to me how can I intuitively rebutle the idea that $x$ (and $y$) are uniformly distributed, without involving calculation?

Answer 1

It is because the circle is more steep on its left and right edges so a small interval there will contain a lot of arc length. But as you go toward the center of the circle, a small interval will be smaller. Consider an interval of [-1,-.99] vs an interval of [0,.01]. The slope is much higher in the first interval than the second, which is mostly flat, so the first interval will contain more “points,” and hence probability.

Also, Compare the interval [-1,-1/2] vs [-1/2, 0], dividing the left half in two equal interval widths. The first interval contains more arc length than the second. To convince yourself, draw it out. It looks like the first contains about twice as much as the second.

The more arc length, the more “points” there are to choose from.

Two intervals of the same width won’t contain the same arc length unless they are mirrors across the y axis.