How to find de Mean and Variance of a defined distribution

Question

I would like to find the Mean and Variance expressions for a Poisson-Lognormal Distribution

$$f(x;\mu,\sigma)=\frac{1}{x!\sigma\sqrt{2\pi}}\int_{0}^{\infty}\lambda^{x-1} e^{-\lambda} e^{\frac{(log(\lambda-\mu)^2}{2\sigma^2} }\text{d}\lambda$$

I have been checking the Expectation[] command to find the expected value of predefined Mathematica pdfs but I do not know how to apply to this expression, and to find the $Var(X)=E[X^2]-E[X]^2$.

Answer 1

I suspect that you will need to use numeric techniques.

Clear["Global`*"]

f[x_Integer?NonNegative,
  μ_?(NumericQ[#] && Element[#, Reals] &),
  σ_?(NumericQ[#] && # > 0 &)] := NIntegrate[
   E^-λ λ^(x - 
       1) E^(-(Log[λ] - μ)^2/(2 σ^2)),
   {λ, 0, ∞}]/(Sqrt[2 π] σ x!)

mean[μ_?(NumericQ[#] && Element[#, Reals] &),
  σ_?(NumericQ[#] && # > 0 &)] := mean[μ, σ] =
  NSum[x*f[x, μ, σ], {x, 0, ∞}, NSumTerms -> 50, 
   WorkingPrecision -> $MachinePrecision]

var[μ_?(NumericQ[#] && Element[#, Reals] &),
  σ_?(NumericQ[#] && # > 0 &)] := var[μ, σ] =
  Module[{m = mean[μ, σ]},
   NSum[(x - m)^2*f[x, μ, σ], {x, 0, ∞}, NSumTerms -> 50,
     WorkingPrecision -> $MachinePrecision]]

mean[0, 1]

(* 1.6487197 *)

var[0, 1]

(* 6.318 *)