Matlab or Mathematica: Need help in computing the Pdf for sum of unknown named distribution and normal random variable

Question

Is it possible to apply convolution theorem or software like Mathematica to find a closed form expression for the pdf of Z = R + X where f_R(r;k,d) = kdr^(d-1)(1-r^d)^(k-1) and X is zero mean Gaussian rv of unknown variance. r ~ [0,1] and the pdf f_R(r;k,d) is related to the probability of drawing one point with distance r multiplied by that of drawing k-1 points with distance > r .

I don't know how to specify an unknown distribution in Mathematica or Matlab if it needs to be used to calculate closed form expressions in cases where analytically it is difficult / impossible.

In Mathematica, we can use existing named distribution like NormalDistribution[mu, std] but how to use f_R(r;k,d) ?

Answer 1

If I'm correct, for k and d positive integers, the convolution integral can be expressed in terms of moments of the standard normal distribution, which are known (see for example here ).

Let f(r) denote the standard normal pdf, and let h(r) denote the other pdf in your problem,

.

Expanding the term (1- r ^d ) ^{k -1} with the binomial theorem , g ( r ) can be expressed as a sum of terms of the form br ^s , where s is integer if k and d are. Let the convolution of f and g be denoted as h :

This integral can be expressed as a sum of terms of the form

times a constant (by "constant" I mean a term that does not depend on the integration variable, and thus can be moved out of the integral). Again expanding ( r - t ) ^s gives terms of the form r ^m · t ⁿ . So the integral can be expressed as a sum of terms

times a constant. These terms are given by the moments of the normal distribution.