Convolution of difference distribution doesn't conserve expectation - faulty assumption? I have observations of a system which I hypothesize is defined as $T = A + G$ where $T,A, G$ are random variables. I've measured the empirical continuous distributions $T$ and $A$, and the question is, what is the distribution of $G$?
Assuming $A$ and $G$ are independent, this is just the convolution integral for $T-G$, that is, since $i-(i-k)=k$,
$$
P_G(G=k)=\int_{i=0}^\infty P_T(T=i)P_A(A=i-k)
$$
Because the data for $T$ and $A$ are empirical distributions, I computed discrete PMFs with various discretizations and the following results don't depend much on what resolution to discretize.
Inputting my data, I get a nice distribution for $G$, however, there are two things I don't understand. 


*

*While the discrete PMFs that approximate $A$ and $T$ sum to 1, the sum of $P_G =.543$. Do I just renormalize?

*$E[G]=.78$ while $E[T]=1.52$ and $E[A]=1.32$. Mean of $G$ is not affected by (1). I know even for correlated variables, $E[A+B]=E[A]+E[B]$, so something must be wrong. The only assumption I've made is that $A$ and $G$ are not correlated (so the convolution applies). Is this result telling me $A$ and $G$ are correlated? What can I do to find out if they are, given what I have? I cannot observe $G$ without $A$.
Edit: changed "Assuming $A$ and $G$ are not correlated" to "Assuming $A$ and $G$ are independent" per Dilip since the terminology used was incorrect.
 A: Your statement

Assuming $A$ and $G$ are not correlated, this is just the convolution integral 

is false. The distribution of the sum or difference of independent
random variables can be obtained via convolution-type formulas, but
this formula does not hold for uncorrelated random variables.
The key issue here is

I've measured the empirical continuous distributions $T$ and $A$ 

whereas what you need is the (empirical) joint distribution of $T$ and $A$;
the individual (a.k.a. marginal) distributions do not suffice.

The only assumption I've made is that A and G are not correlated (so the convolution applies). Is this result telling me A and G are correlated?

What the result is telling you is that $A$ and $G$ are dependent random variables. They might be correlated, and they might not be correlated;
it all depends on the joint distribution.  For example, if $(A,G)$
is uniformly distributed on the unit disc, then $A$ and $G$ are 
uncorrelated but dependent random variables.
Applying your convolution formula to the sum/difference of $A$ and $G$
in this case will not give you the distribution of $A\pm G$.
