Suppose we have independent random variables $X_1,\ldots,X_n$ with the same distribution. Let $F(\cdot)$ be the cumulative distribution of $X_i$ for all i We are interested in the cumulative distribution of (Max,Min) where $\text{Max}=\max\{X_1,\ldots,X_n\}$ and similarly for min. This is what I've worked out for far;
Let $G(x,y)$ the cumulative distribution of (Max,Min)
\begin{align} G(x,y) & = P(\text{Max}\leq x, \text{Min} \leq y) \\[8pt] & = P(\text{Max}\leq x) - P(\text{Max} \leq x, \text{Min} \gt y ) \\[8pt] & = P(X_1 \leq x,\ldots,X_N \leq x) + P(y \lt X_1 \leq x) \\[8pt] & = F^n(x) - [F(x) - F[y]]^n \end{align}
Is this correct? Thanks