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:

  with  

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

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

  with  

With the following alternative auxilary variable, the solution becomes

  with     ()

http://web.ist.utl.pt/~mcasquilho/acad/theo/simul/parabolic.php Created: 2006-03-16 — Last modified: 2018-11-18