Problem with TransformedDistribution

Question

I am trying to use Mathematica to obtain the probability distribution of $\frac{1}{2}(A + B)$ where $A$ and $B$ are independent random variables each distributed according to the uniform distribution, with lower and upper bounds of $L$ and $H$ respectively.

I suspect the distribution is triangular with lower and upper bounds of $L$ and $H$ respectively and mode equal to $\frac{1}{2}(A + B)$. However, I am having difficulty using TransformedDistribution to show that.

My code is:

\[ScriptCapitalD] = TransformedDistribution[1/2 (A + B), {B \[Distributed] UniformDistribution[{L, H}], A \[Distributed] UniformDistribution[{L, H}]}]

Answer 1

You get what you expect if you do it it in two steps

\[ScriptCapitalD] = 
 TransformedDistribution[x/2, 
  x \[Distributed] TransformedDistribution[(A + B), {
     B \[Distributed] UniformDistribution[{L, H}],
     A \[Distributed] UniformDistribution[{L, H}]}]]

(* TriangularDistribution[{L, H}] *)

Answer 2

PDF[\[ScriptCapitalD]][z]

(((-30 + z)Sign[-30 + z])/2 - (-20 + z) Sign[-20 + z] + ((-10 + z)*Sign[-10 + z])/2)/100

For plotting, assign values to L and H:

L = 10; H = 30;
Plot[Evaluate@PDF[\[ScriptCapitalD]][x], {x, 10, 30}]

pdF[l_, h_] := Module[{L = l, H = h}, Evaluate[PDF[\[ScriptCapitalD]]]]
Plot[Evaluate @ Flatten@Table[pdF[l, h][x], {l, {0, 5}}, {h, {10, 15}}], {x, 0, 15},
  PlotRange -> All, 
 PlotLegends -> (Flatten @ Table[ToString@{l, h}, {l, {0, 5}}, {h, {10, 15}}])]