Overloading functions like Mean for distributions

Question

I'd like to understand how functions like Mean are overloaded to provide the right behavior for the distributions in Mathematica (like NormalDistribution or PoissonDistribution).

I originally assumed it was through the use of UpValues, but now I'm not so sure...

So, if I wanted to implement a function or distribution and define a behavior for it when another function like Mean is applied to it, how would I go about it? I know it's not the following:

f[x_]:=2+x
f /: Mean[f[x_]] := 3 x

Desired behavior:

f[3]
Mean[f[3]]

(*
==> 5
==> 9
*)

Answer 1

Symbolic(!) distributions are recognized by their head, not by their PDF:

Mean[PDF[NormalDistribution[0, 1], x]] 

Mean[E^(-(x^2/2))/Sqrt[2 π]]

So instead of messing around with Mean, I would rather suggest

distro /: PDF[distro[μ_, σ_], x_] := E^(-((x - μ)^2/(2 σ^2)))/(Sqrt[2 π] σ);
distro /: Mean[distro[μ_, σ_]] := μ;
distro /: Variance[distro[μ_, σ_]] := σ^2
Mean[distro[0, 1]]
Variance[distro[0, 1]]

0

1

Answer 2

This works:

Unprotect[Mean];
SetAttributes[Mean, HoldFirst];
Protect[Mean];
f[x_] := 2 + x
f /: Mean[f[x_]] := 3 x

Since using Unprotect is not reccomended here's another way.

mean[x_] := Mean[x]
SetAttributes[mean, HoldFirst]
f[x_] := 2 + x
f /: mean[f[x_]] := 3 x