What is the PDF of random variable Z=XY?

If $X$ and $Y$ are two random variables, you may find the product distribution as follows: $$ f_Z(z)=\int_{-\infty}^{\infty} \frac{1}{|t|} f_X\left(t\right)f_Y\left(\frac{z}{t}\right)dt $$ To see this, suppose that the distribution of $X$ is continuous at 0: $$ P(Z\leq z)=P(XY\leq z)= P(Y\leq \frac{z}{X}\big|X>0)P(X>0)+P(Y\geq \frac{z}{X}\big|X<0)P(X<0)= $$ $$ =\int_{0}^{\infty} P(Y\leq \frac{z}{t}\big)f_X\left(t\right)dt+ \int_{-\infty}^{0} P(Y\geq \frac{z}{t}\big) f_X\left(t\right)dt $$ We can find the derivation of both sides w.r.t. $z$ and we get:

$$ f_Z(z)=\int_{0}^{\infty} \frac{1}{t} f_Y(\frac{z}{t}\big)f_X\left(t\right)dt+ \int_{-\infty}^{0} \frac{-1}{t} f_Y(\frac{z}{t}\big) f_X\left(t\right)dt $$ $$ =\int_{-\infty}^{\infty} \frac{1}{|t|} f_X\left(t\right)f_Y\left(\frac{z}{t}\right)dt $$ The moral: it is not always possible to use this formula.