Random numbers are generated according to the standard normal distribution X ~ N(0,1). They are shown on the x-axis of figure (a). Many of them are around 0. Each of those numbers is mapped according to y = x², which is shown with grey arrows for two example points. For many generated realisations of X, a histogram on the y-axis will converge towards the wanted probability density function rho_Y shown in (c).
In order to analytically derive this function, we start by observing that in order for Y to be between any v and v+Dv, X must have been between either -sqrt(v+Dv) and -sqrt(v), or, between sqrt(v) and sqrt(v+Dv). Those intervals are marked in figures (b) and (c). Because the probability is equal to the area under the probability density function, we can determine rho_Y from the condition that the grey shaded area in (c) must be equal to the sum of the areas in (b). The areas are calculated using integrals and it is useful to take the limit Dv -&gt; 0 in order to get the formula noted in (c).
Own work
LarsWinterfeld