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In the context of a numerical simulation I have normally distributed random variables $A$ and $C$. These variables are based on real data. I know that there is a distribution $B$ where $A + B = C$.

Let $A = N($$\mu_a$,$\sigma_a$)

Let $C = N($$\mu_c$,$\sigma_c$)

$\sigma_c > \sigma_a$

If I only know $C$ and $A$, how could I go about calculating $B$ using some numerical method?

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  • $\begingroup$ Umm, $B = C - A$. Or do you mean you want the distribution of $B$? $\endgroup$ – Robert Israel Sep 19 '16 at 19:23
  • $\begingroup$ Yes, I want the distribution of $B$ $\endgroup$ – Hefaestion Sep 19 '16 at 19:25
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If all you know are the distributions of $A$ and $C$, you can't determine the distribution of $B$ without some other assumptions. For example, if $A$ and $B$ are assumed independent, then $B \sim N(\mu_c - \mu_b, \sqrt{\sigma_c^2 - \sigma_a^2})$. But if $A$ and $C$ are assumed independent, then $B \sim N(\mu_c - \mu_b, \sqrt{\sigma_c^2 + \sigma_a^2})$. And there are also examples where $B$ does not have a normal distribution at all (see e.g. here).

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