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Possible Duplicate:
Proof that the sum of two Gaussian variables is another Gaussian

Let $X,Y$ be independent normally distributed $N(0,1)$ random variable, and $\alpha,\beta\in \mathbb{R}$. What is the cumulative distribution function of $\alpha X+\beta Y$?

Thank you very much for your help.

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    so I have like $\int_{\infty} ^{\infty} \frac{1}{2\pi} \frac{1}{\alpha\beta} e^\frac{(-x/\alpha)^2}{\alpha} e^\frac{-((z-x)/\alpha)^2}{\alpha}dx$ but i have no idea how to calculate it2011-09-18
  • 0
    The integral can be reduced via algebra to a well-known integral. You need to know when and how to complete a square.2011-09-18
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    This *must* be a duplicate...2011-09-18
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    That page lacks a proper proof and the only answer points to the wikipedia page for a proof. Perhaps someone should post another more descriptive answer there? Or, is the wikipedia link enough? (The article seems pretty well-written and complete.)2011-09-18

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