Let $X_1$ have p.d.f $$p_1(x_1)=x_1 \cdot \text{exp} \left( \frac{-x_1^2}{2} \right),$$ and $X_2$ have p.d.f $$p_2(x_2) = \frac{1}{\sqrt{2 \pi}} \text{exp} \left( \frac{-x_2^2}{2} \right). $$ Calculate the distribution of $$X=X_1X_2.$$
In the solution it is claimed that $$p(x) = \iint_{\mathbb{R}^2} p_1(x_1)p_2(x_2) \delta(x-x_1x_2) \, dx_1dx_2=...,$$ where $\delta$ is the dirac-delta function. I have two questions:
- Can someone give me the intuition behind this formula?
- Does this hold independent of $X_1$ and $X_2$ beeing independent?