# Thread: Uniform Distribution of a Circle

1. ## Uniform Distribution of a Circle

Suppose that $\displaystyle (X_i, Y_i), i = 1,...,n$ is a 2D sample, with each $\displaystyle (X_i, Y_i)$ assumed to be uniformly distributed in a circle centered at the origin of an unknown radius $\displaystyle r$.

How do I calculate the pdf of this?

I'm thinking $\displaystyle \frac{1}{\pi r^2}$ but I don't think this is correct.

2. Originally Posted by BERRY
Suppose that $\displaystyle (X_i, Y_i), i = 1,...,n$ is a 2D sample, with each $\displaystyle (X_i, Y_i)$ assumed to be uniformly distributed in a circle centered at the origin of an unknown radius $\displaystyle r$.

How do I calculate the pdf of this?

I'm thinking $\displaystyle \frac{1}{\pi r^2}$ but I don't think this is correct.
The uniform distribution in the disc of radius $\displaystyle r$ has indeed the following p.d.f.: $\displaystyle \frac{1_{(x^2+y^2\leq r^2)}}{\pi r^2}$.

(sometimes it is more useful to have an expression in terms of polar coordinates)