# Deriving the characteristic function for $N(0,2)$

Could someone please help me with an easy derivation of the characteristic function for a $N(0,2)$ distribution? Or a link to somewhere it is done.

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## 1 Answer

• Once you know the characteristic function of $N(0,1)$, you can deduce the corresponding for $N(m,\sigma^2)$ for each $m$ and $\sigma$.
• Let $\varphi$ the characteristic function of $N(0,1)$. We have $\varphi(t)=\frac 1{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{itx}e^{-x^2/2}dx$. We can take the derivative and find a differential equation satisfied by $\varphi$. Using the initial condition $\varphi(0)=1$ you can completely determine $\varphi$.
An other method is to write $\varphi(t)=\frac 1{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{-1/2(x-it)^2}dx$ and use a contour integral.
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Could I do it in any way without assuming N(0,1)? Actually deriving that would maybe be sufficient. – Henrik Jun 9 '12 at 13:33
It will involve the same tools. – Davide Giraudo Jun 9 '12 at 13:34
Thanks! I have derived it in that way then, I just hoped that a way not involving differential equations existed. – Henrik Jun 9 '12 at 13:56