Distribution of the Maximum and Minimum of a Random Number of Bounded Random Variables

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 $X_1,X_2,\dots ,X_N$, 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.