Mean of TransformedDistribution returns wrong result

Question

I want to compute the mean of this transformed distribution $10^{(25+5x+y)/5}$, where $x$ and $y$ are independent normal random variables.

Here is my code

oneStepDist = 
  TransformedDistribution[10^(
   1/5 (25 + 5 x + y)), {x \[Distributed] 
     NormalDistribution[71273/100000, 11/6250], 
    y \[Distributed] NormalDistribution[-4811/250, 21/500]}];
twoStepDist = 
  TransformedDistribution[10^t, t \[Distributed] #] &@
   TransformedDistribution[
    1/5 (25 + 5 x + y), {x \[Distributed] 
      NormalDistribution[71273/100000, 11/6250], 
     y \[Distributed] NormalDistribution[-4811/250, 21/500]}];

I found Mean[oneStepDist] returns $0$ while N@Mean[twoStepDist] gives correct result $73.1164$. Is this a bug or I'm missing something?

Answer 1

Just an extended comment stating that the one-step can be made to work but it's a very fragile approach. The list z below is a list of equivalent formulations of the desired random variable:

z = {10^(1/5 (25 + 5 x + y)), Exp[5 Log[10] + Log[10] x + Log[10] y/5], 
  Exp[Log[10] (5 + x + y/5)], Exp[a (5 + x + y/5)]};
TableForm[results = {#, Mean[TransformedDistribution[#, {x \[Distributed] 
   NormalDistribution[71273/100000, 11/6250], 
   y \[Distributed] NormalDistribution[-(4811/250), 21/500]}]]/. a -> Log[10]} & /@ z, 
 TableHeadings -> {None, {"Random variable", "Mean"}}]

TableForm[results // N, TableHeadings -> {None, {"Random variable", "Mean"}}]

The moral is to avoid random variables of the form 10^z ??