How to calculate the expectation and variance of $\cos(X)$,where X obeys Standard Normal Distribution?

Question

Calculate the expectation and variance of $\cos(X)$,where $X$ obeys Standard Normal Distribution.

In my homework, I encountered the problem. At first glance, I thought it was just a simple integration. But soon I found that I couldn't figure out the integration. So I am wondering if there is any technique I can apply to this integration or if there is an easier path to solve this problem.

Welcome to MSE. A question should be written in such a way that it can be understood even by someone who did not read its title. — José Carlos Santos, Aug 30 '22 at 07:32

score 4 · Accepted Answer · answered Aug 30 '22 at 07:23

Hints: We can do the integration by using Euler's formula:$cos(x)=\frac{1}{2}(e^{ix}+e^{-ix})$. For an easier path, you can also use Euler's formula $E(cos(X))=\frac{1}{2}(E(e^{iX})+E(e^{-iX}))$ and note that the terms are exactly in the form of the characteristic function of X.

How to calculate the expectation and variance of $\cos(X)$,where X obeys Standard Normal Distribution?

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