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.
Deriving the characteristic function for $N(0,2)$
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probability-theory
normal-distribution
characteristic-functions
1 Answers
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- 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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0Thanks! I have derived it in that way then, I just hoped that a way not involving differential equations existed. – 2012-06-09