Dear all,


I have recently been breaking my head over this question. The idea is that a certain variable $Y$ is normally distributed with a parameter $X$ in both mean and variance. 


$Y|X \sim N(\mu X,X^2)$


This parameter $X$ is assumed to be normally distributed as well with parameters $\alpha$ and $\beta$. 


$X\sim N(\alpha, \beta)$


Now I am interested in the distribution of $Y$ (with $X$ marginalized out). 


My current progress:
Simulation shows me that the distribution of Y seems approximately normal as well. This does not proof anything off course. 


The integral $\int\_{all x}f\_{Y|X}(s;x)f\_{X}(s)dx$ seems unsolveable. 


Kind regards



$Y|X \sim N(\mu X,X^2)$ and $X\sim N(\alpha, \beta)$. How is $Y$ distributed?
The characteristic function is 
$$\eqalign{ E\left[e^{itY}\right] &= E\left[ E\left[ e^{itY}|X \right]\right] \cr
&= E \left[ \exp(it\mu X - t^2 X^2/2 \right] \cr
&= \frac{\exp \left(\left(-(\alpha^2+\beta \mu^2) t^2 + 2 i \mu \alpha t\right)/\left(2 t^2 \beta + 2\right)
\right)}{\sqrt {{t}^{2}
\beta+1}}\cr}$$


It is certainly not a normal distribution, but might be approximated by a normal distribution when $\beta$ is small and $\alpha \ne 0$. In fact, by expanding this in a series in powers of $\beta$ and taking the inverse Fourier transform, I get a density


$$f(y) = \frac{\exp(-(y-\mu \alpha)^2/(2 \alpha^2))}{\sqrt{2 \pi |\alpha|}} \left(
1 + 
\frac { \left( 2\;{\alpha}^{4}+4\;x{\alpha}^{3}\mu+{x}^{2} \left( -5+{\mu}^{2} \right) {\alpha}^{2}-2\;{x}^{3}\alpha\mu+{x}^{4
} \right) }{2{\alpha}^{6}} \beta + \ldots\right)
$$


