Suppose that I have 2 independently distributed normal random variables, $A \sim N(\mu_A, \sigma^2_A)$ and $B \sim N(\mu_B, \sigma^2_B)$. I do not observe either of these variables. What I observe is their sum, $C=A+B$. Given a value of the sum, $C=c$, what is the conditional distribution of $A$ (or $B$)? I am hoping that there is a closed form analytical solution for this question which is a function of the 5 parameters: $\mu_A, \sigma^2_A,\mu_B, \sigma^2_B, c$.
Also, if it is possible, what I am really interested in is the case where $A$ and $B$ are log-normally distributed. But from what I can tell working with sums of lognormal variables is complicated. So an answer for the case of normality would suffice.