Simulation of a "parabolic" variable

Suppose a parabola as a pdf (similar to the central part of the "bell" in a Gaussian variable). Then, it is:

pdf   with   pdf_dom

This leads by integration to     (See the graphs.xls)

cdf

which is a 3.rd degree polynomial. The resolution of this cubic —to obtain random x's from given Y 's— becomes relatively simple due to the absence of the term of the 2.nd degree, being a "case 7" (in the reference mentioned, since it is b = 0) for all the values of Y ("case 6" in the extremes). Thus, with the following auxiliary variable, the solution is

x_of_v   with   aux_v

With the following alternative auxilary variable, the solution becomes

x_of_beta   with   aux_beta   (beta_dom)
An illustrative Excel file
 
 
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Created: 2006-03-16 — Last modified: 2018-11-18