Random variables, X and Y, are uniformly distributed in the
quarter circle, with center at the origin and a radius of one
in the first quadrant of the x,y plane. Please find the joint PDF of X and Y.
Attempt at Solution:
I've worked a problem similar to this one before, except instead of a quarter circle, it involved a full circle. In such a problem, the equation for the distribution was $1/(\pi * r^{2})$, which makes sense as this follows the trend apparent in any continuous uniform distribution I have seen. However, in this case, since the area of the circle is $(pi * r^{2})/4$, the inverse of this, $4/\pi$, would yield probability density greater than one.
So have I approached the problem correctly (i.e. is it ok to have a probability density in excess of one), or is there something that I am missing here?
Best Answer
Indeed the joint PDF of $(X,Y)$ is $$f_{(X,Y)}=\frac4\pi\,\mathbf 1_D,$$ where $D$ is the quarter disk you described. Note that PDFs with values larger than $1$ are common, for example $Z$ uniform on the interval $K=(0,1/10)$ has PDF $$f_Z=10\cdot\mathbf 1_K.$$