Characteristic function proof

Question

How to prove that: $$ \frac{\exp(-x^2)+1}{2}$$ is a characteristic function?

I will be grateful for any help!

Answer 1

Making explicit the my comment above.

We know that if $X\sim N(\mu,\sigma)$ then $\phi_X(t)=e^{i\mu t-\frac{1}{2}\sigma^2t^2}$

Then $e^{-x^2}$ is the characteristic function of a $Y\sim N(0,\sqrt{2})$.

Notice that it $Z$ is constant $=0$, then $\phi_Z(t)=E[e^{itZ}]=E[1]=1$. The distribution (density!? I don't know well the names in probability theory) of such a $Z$ is the Dirac delta at the origin, $\delta_0$.

Therefore

$$\begin{align}\frac{e^{-t^2}-1}{2}&=\frac{E[e^{itY}]+E[e^{itZ}]}{2}\\&=\frac{\int e^{its}\frac{e^{-s^2/4}}{2\sqrt{\pi}}\text{d}s+\int e^{its}\delta_0(s)\text{d}s}{2}\\&=\int e^{its}\left(\frac{e^{-s^2/4}}{8\sqrt{\pi}}+\frac{\delta_0(s)}{2}\right)\text{d}s\end{align}$$

Therefore $\frac{e^{-x^2}+1}{2}$ is the characteristic of a random variable with density function

$$\frac{e^{-s^2/4}}{8\sqrt{\pi}}+\frac{\delta_0(s)}{2}$$

(If I didn't make a computation mistake).