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Given the two variables $A\sim \text{N}(\mu, \phi^2)$ and $B\sim \text{N}(\xi, \omega^2)$ with $\mu, \xi \in R$ and $\phi^2, \omega^2 > 0$ how do I prove that $C := A + B\sim \text{N}(\mu + \xi, \phi^2 + \omega^2)$ ?

SimpleCalc
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  • You want the normality of the sum of independent normals. See http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variable. – deinst Jul 12 '14 at 14:44
  • @deinst An "s" is missing in the link. http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables – callculus42 Jul 12 '14 at 14:52
  • Please search using the keywords from your question (e.g., sum, independent, normal) before posting it. See [ask]. –  Jul 12 '14 at 15:05

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in case they are independent use Characteristic functions: $$\phi_{A+B} (t) = \phi_{A}(t) \phi_{B}(t)$$ and it is easy to see that $\phi_{A}(t)$ looks like char. function. of $N(\mu+\xi, \omega^{2}+\phi^{2})$

Kanye West
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