How to prove the characteristic function of normal distribution?

Question

How to prove that the characteristic function of the normal distribution $N(a, σ^2)$ has the form: $$φ (t) = e^{ita - \frac{1}2t^2σ^2}$$

I think that we need to use this formula, but I don't know what to do next: image

Answer 1

You can use the relation:

$$\varphi(t)=\int_{-\infty}^{+\infty} p(x)e^{ixt}\text{d}x$$

where $p$ is the normal density probability function:

$$p(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left[\frac{(x-\mu)^2}{2\sigma^2}\right]$$

So you have:

$$\varphi(t)=\frac{1}{\sqrt{2\pi\sigma^2}}\int_{\mathbb{R}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}e^{ixt}\text{d}x=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{\mu^2}{2\sigma^2}}\int_{\mathbb{R}} e^{-\frac{x^2}{2\sigma^2}+\left(\frac{\mu}{\sigma^2}+it\right)x}\text{d}x= \\ \\ =\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{\mu^2}{2\sigma^2}}\sqrt{2\pi\sigma^2}e^{\left(\frac{\mu}{\sigma^2}+it\right)^2\frac{\sigma^2}{2}}=e^{-\frac{t^2\sigma^2}{2}+i\mu t}$$

I have used the gaussian integral:

$$\int_{\mathbb{R}} e^{-Ax^2+Bx}\text{d}x=\sqrt{\frac{\pi}{A}}e^{B^2/4A}, \quad A>0, \quad B \in \mathbb{C}$$