Let $Z=X \cos(Y)$ be a random variable where $X\sim\mathcal N(0,\sigma_1)$, $Y\sim\mathcal N(0,\sigma_2)$. $X$ and $Y$ are independent.My goal it to obtain the PDF of Z or at least prove some behaviors.
My first approach was to evaluate the pdf of $\cos (Y)$ and after that, use the convolution theorem to get $Z$. However, I think that it's not a good way because the pdf of $\cos (Y)$ is a mess]1.
After lost time with this approach, I investigated pdf of $Z$ numerically.
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2]
This distribution resembles a Laplace (exponential) distribution!
Because of the exponential decay behavior (figure), I tried to discover the characteristic function of $Z$. The reason to do that, it’s because if the characteristic function of $Z$ coincides with the characteristic function of the Laplace distribution then I solve my problem.
I obtained the following integral for the characteristic function of $Z$, $$ \varphi_Z(t) = \frac{1}{\sqrt{2\pi}\sigma_2} \int\limits_{-\infty}^\infty \exp\left(-\frac{(t \sigma_1 \cos(y))^2}{2}\right) \exp\left(-\frac{y^2}{2\sigma_2^2}\right) \mathrm{d} y, $$ I tried to solve that integral expanding the first exponential.
$$ \varphi_Z(t) = \sum_{n=0}^\infty\frac{(-1)^n}{n!} \left(\frac{\sigma_1 t}{\sqrt{2}}\right)^{2n} \frac{1}{\sqrt{2\pi}} \sum_{k=0}^{2n} \binom{2n}{k} \exp(-2\sigma_2^2(n-k)^2). $$ However, I don’t know if I’m doing something wrong. This equation doesn’t look useful.
My questions are
Is there another way to calculate the pdf of $Z$?
How to prove the exponential decaying behavior of pdf $Z$ without knowing the pdf of $Z$?