I have uniformly distributed nodes over an annular region RI<r<RO. I would like to compute the PDF of the distance from any chosen node to the center of the annular. In the case of $R_I = 0$ (circle), the PDF is simply given by $r^2 / R_O^2$ as solved here for $R_I = 0$ The logic in my case is to have PDF = $r^2 - (R_I)^2 / (R_O^2 - R_I^2)$. However, when I simulated it in MATLAB, this formula doesn't match with the simulation!
Could anyone help, please?
Note that the pdf given for $\ R_I\ $ in the answer you cite is $\ \frac{2r}{R_O^2}\ $, not $\ \frac{r^2}{R_O^2}\ $, as you say here. The latter is the cumulative distribution function of the point's distance from the origin. For $\ R_I\ne0\ $ the cumulative distribution function of the point's distance from the origin is $$ \frac{r^2-RI^2}{RO^2-RI^2}\ . $$ for $\ RI\le r\le RO\ $, and the pdf is the derivative of this: $$ \frac{2r}{RO^2-RI^2}\ . $$