Suppose $X_i\sim f_{X_i}(x)$, with each $X_i$ independent and identically distributed for some arbitrary probability density function $f_{X_i}(x)$. Given this information, and this information only, define the new random variable:
$$Y=\prod_{i=1}^nX_i$$
What is the distribution of $Y$? Particularly, what is its probability density function? If each $X_i$ is a discrete random variable, what's the analog? If there's no easy way to obtain $f_Y(y)$, is there a way to get the moment generating function of $Y$ instead?