I am trying to obtain the pdf of a random variable with a Gaussian distribution $X \sim \mathcal{N}(0, A)$, whose variance A is determined by another random variable A with an exponential distribution with $\lambda=1$.
So the process would be the following:
- Obtain a particular value $a$ from random variable $A$
- Obtain a particular value $x$ from random variable $X$ with variance $a$
I have found that the correct pdf for random variable X is:
$$ \frac{1}{\sqrt{2}}\exp\left(-\sqrt{2}|x|\right) $$
And I have also checked that this formula is correct doing a simulation. But, how can I compute this pdf analitically?
Simulated histogram + plotted pdf
Thanks!