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

Question

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.

Answer 1

• 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.