Distribution of the product of a Gaussian Random Variable with a scaled and mean shifted value of itself

Question

Suppose we have a r.v. $X$ with pdf $N(0,\sigma^2)$ if I get another r.v. formed by

$$ y=\alpha + \beta \times x$$

What is the PDF of

$$ Z = x \times y $$

To be precise, how to get the joint distribution if they are fully correlated?

Answer 1

$Z = \alpha X + \beta X^2$. We have to be careful because this is not a one-to-one function of $X$. Let's suppose e.g. that $\beta > 0$. Then the minimum possible value of $\alpha x + \beta x^2$ is $-a^2/(4b)$. For $z > -a^2/(4b)$,

$$\eqalign{\mathbb P(Z \le z) &= \mathbb P\left( \dfrac{-a - \sqrt{a^2 + 4 b z}}{2b} \le X \le \dfrac{-a + \sqrt{a^2 + 4 b z}}{2b} \right)\cr &= \int_{ (-a - \sqrt{a^2 + 4 b z})/(2b)}^{(-a + \sqrt{a^2 + 4 b z})/(2b)} f(x)\; dx}$$

Differentiate using the Fundamental Theorem of Calculus to get the density.