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I have to find the random variable $X$ s.t. $\psi(t)=\cos(t)$ is its characteristic function.

I'm trying to use the inversion formula $f(x)=\int_{-\infty}^{+\infty} e^{-itx}\psi(t)dt$ but it seems to get me nowhere.

Is there some kind of trick on this? I also tried to find the distribution making a sequence $\psi_n \to \psi$ from the Taylor expansion of $\cos(t)$ but also nothing.

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    What is the characteristic function of a discrete random variable that takes on values $+1$ and $-1$ with equal probability $\frac 12$? – 2017-01-08
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    @DilipSarwate Thank you! Seems kind of obvious now you mention it. – 2017-01-08

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