Matlab or Mathematica: need help calculating Pdf for the sum of an unknown name and an ordinary random variable

Question

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

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

Answer 1

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

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

.

(1-r ^d) ^k-1 , g (r) br ^s s , k d. f g h:

( "" , , , ). (r-t) ^s r ^m t ⁿ. ,

. .