English: Figure (a) shows a sketch of random sample points from a uniform distribution in a circle of radius 1, subdivided into rings. In figure (b), we count how many of those points fell into each of the rings with the same width Dv. Since the area of the rings increases linearly with the radius, one can expect more points for larger radii. For Dv -> 0, the normalized histogram in (b) will converge to the wanted probability density function rho_Y. In order to calculate rho_Y analytically, we first derive the cumulated distribution function F_Y, plotted in figure (d). F_Y(y) is the probability to find a point inside the circle of radius v (shown in grey in figure (c) ). For v between 0 and 1, we find F_Y(y) = Av / Atot = πv² / π1² = v². The slope of F_Y is the wanted probability density function rho_Y(y) = dF_Y(y) / dy = 2v, in agreement with figure (b).