I'm having trouble with the following random variable transformation:
$Y = X^2 + X$
I am looking for the pdf of Y. I tried the following method:
$p_Y(y) = \int_{X} p_{Y|X=x}(y)\cdot p_{X}(x)dx$ and we know that $(Y|X=x) \sim (x^2+x) \Rightarrow p_{Y|X=x} = \delta_{x^2+x}(y)$ thus: $p_Y(y) = \int_{X} \delta_{x^2+x}(y)\cdot p_{X}(x)dx$
But I don't see a way to reduce this further.
Then I tried a different approach:
$p_Y(y) = DF(Y < y) = DF(X^2 + X < y) = ...$
But then I don't see a way to find the inverse of $X^2 + X$.
Can anyone help me further on this?