  We study a new family of random variables, that each arise as the
distribution of the maximum or minimum of a random number $N$ of i.i.d.~random
variables $X_1,X_2,\ldots,X_N$, each distributed as a variable $X$ with support
on $[0,1]$. The general scheme is first outlined, and several special cases are
studied in detail. Wherever appropriate, we find estimates of the parameter
$\theta$ in the one-parameter family in question.
