Suppose I have two random variables:
$X_1$ and $X_2$ have densities $f_\theta = \theta x^{\theta - 1}$
I want to find pdf of $Y = X_1 X_2$
$$ P(Y\leq y)= P(X_1X_2\leq y) = \int_0^1 \theta P(X_1\leq y/x) x^{\theta - 1} \, dx = \int_0^1 \left(\frac y x \right)^\theta x^{\theta -1} \, dx $$
Density is derivative, so:
$$\int_0^1 -\theta x^{\theta - 2} \left(\frac{y}{x}\right)^\theta \, dx$$
Is it correct calculations? If not, please tell me, where did I do a mistake