# Normalize an L(s, d) function for fixed s

I have the following function L:

σ = 9;
L[s_, d_] = (1/(σ*Sqrt[2*\[Pi]]))*E^(-(1/2)*((s-(-50-11*Log[d]))/σ)^2)

This function L describes basically for some specific d, the distribution of the random variable s. Therefore, for some fixed d value, the L[s] has the properties of a PDF. What I would like to do is find another function R[s_, d_] that does the opposite. That means, that for a fixed s value, the R[d] is the normalization version (normalized between [dmin, dmax]) of the corresponding L[d].

Any idea if this is even possible? Thank you for your time

σ = 9;
L[s_, d_] = (1/(σ*Sqrt[2*π]))*
E^(-(1/2)*((s - (-50 - 11*Log[d]))/σ)^2);

The probability distribution is

dist[s_] =
ProbabilityDistribution[L[s, d], {d, 0, Infinity},
Method -> "Normalize"];

r[s, d] is the PDF of the distribution

r[s_, d_] = PDF[dist[s], d]

Verifying the normalization,

Integrate[r[s, d], {d, 0, Infinity}]

(* 1 *)

The mean of the distribution is

Mean[dist[s]]

(* E^(-(857/242) - s/11) *)

The variance of the distribution is

Variance[dist[s]]

(* (-1 + E^(81/121)) E^(-(857/121) - (2 s)/11) *)

To generate random data from the distribution for a particular value of s:

SeedRandom[1234];

data = RandomVariate[dist[1], 200];

Show[
Histogram[data, Automatic, "PDF"],
Plot[r[1, d], {d, 0, 0.12}]]

• @Gouz This is why you shouldn't accept an answer too quickly. Here is a much better answer than I gave. You can change the accept. – JimB Apr 17 at 3:34

If the $$d$$ takes on values from 0 to $$\infty$$, then the normalizing constant is given by

Integrate[L[s, d], {d, 0, ∞}]
(* 1/11 E^(-(1019/242) - s/11) *)

So you could define the following function which is non-negative and integrates to 1 for a pdf associated with $$d$$:

r[s_, d_] := L[s, d]/(1/11 E^(-(1019/242) - s/11))

You should avoid using capital letters for functions you define.

• Nice. May I ask, is it possible on Mathematica to show the Integration steps/process? – Gouz Apr 16 at 23:18
• Use Rubi in Mathematica: rulebasedintegration.org to show steps. – JimB Apr 16 at 23:39