Let's say I have a Gaussian random variable $X \sim \mathcal{N}(0,1)$ and the random variable defined as
$Y = a X + b X^2$.
How would I go about finding the pdf for $Y$? I know that for the sum of two independent variables, I can do the convolution. But in this case, $X$ and $X^2$ are not independent, so I'm not sure what to do.
Here I have defined instead $Y = aX^2 + bX$ for reasons that will become clear:
$$\Pr[Y \le -c] = \Pr[a X^2 + b X + c \le 0] = \Pr\left[\frac{-b - \sqrt{b^2 - 4ac}}{2a} \le X \le \frac{-b+\sqrt{b^2-4ac}}{2a}\right].$$ Define $$u(c) = \frac{-b + \sqrt{b^2-4ac}}{2a}, \quad l(c) = \frac{-b - \sqrt{b^2-4ac}}{2a}.$$ Then $$F_Y(-c) = \Phi(u(c)) - \Phi(l(c))$$ provided $b^2 - 4ac \ge 0$ or equivalently $-c \ge -b^2/(4a)$; otherwise the probability is $0$. Differentiation gives $$f_Y(c) = f_X(l(-c))l'(-c) - f_X(u(-c))u'(-c)$$ where $f_X$ is the standard normal density.