I want to estimate the pdf distribution of a polynomial transformation of a known continuous random variable $X$:
$Y = f(X) = a_nX^n + \cdots + a_1X + a_0$
Normally, the only numerical approach I know of is by using a simulation. What I tried to do:
evaluate discrete points the pdf of $Y$ by solving a root finding problem:
$P(a_nX^n + \cdots + a_1X + a_0 = c) = P(X = \alpha)$, where $\alpha$ is the root of $a_nX^n + \cdots + a_1X + a_0 - c = 0$.
I do this for varying $c$. I then evaluate these $\alpha$'s in the continuous distribution of the known $X$, and use the found points to fit a curve as approximation to the pdf of $Y$.
When I try this, the shape of the curve is right, however, it is not normalized.
I believe I am making an error by evaluating discrete points in a continuous distribution (which should be 0 by definition), but why is it that the shape of the curve is still right?