Suppose we have a collection of IID random variables \(\{ X_1, \ldots, X_n \}\), and we also have a second collection of IID random variables \(\{ Y_1, \ldots, Y_m \}\). Each \(X_i \sim F_X\) and \(Y_i \sim F_Y\) and we will assume that all these variables are statistically independent. Let us also assume that that \(F_X\) and \(F_Y\) are in the \(\mathcal{C}^1\) smoothness class.
Suppose we would like to find \(Pr \left[ \min (X_1, \ldots, X_n) > \max (Y_1, \ldots, Y_n) \right]\), which is equal to \(Pr \left[ \min (X_1, \ldots, X_n) - \max (Y_1, \ldots, Y_n) > 0 \right]\). The relevance of this observation is that \(\min (X_1, \ldots, X_n) - \max (Y_1, \ldots, Y_n)\) is an expression for which we can derive the distribution
For the minimum of the collection of \(X\) variables we can use order statistics to obtain:
\[F_{X_{(1)}}(x) = Pr \left[ \min \{X_1, \ldots, X_n \} \leq x \right] = 1 - \left[1 - F_X(x) \right]^n.\]
Likewise, the maximum of the \(Y\) variables comes from order statistics:
\[\max (Y_1, \ldots, Y_m) \sim \left[ F_Y \right]^m\]
We would like to put our problem into the form of adding two independent random variables \(U + V\) because then we can convolve them to obtain the distribution of the sum. Taking \(U = X_{(1)}\) as our minimum of the \(X\) variables, and \(V = - Y_{(m)}\) of the \(Y\) variables, we can next consider the distribution of \(V\) to be a reflection of \(Y_{(m)}\). The smooth change in variables works out to be
\[f_V(v) = m f_Y(-y) \left[ F_Y(-y)\right]^{m-1}.\]
To compute the convolution of the densities \(f_U \star f_V\) we need the density \(f_U\):
\[f_U(u) = \frac{d}{dx} F_X(x) = n \left[ 1 - F_X(x) \right]^{n-1}f_X(x)\]
We can use the convolution theorem to obtain the result via the Fourier transform \(\mathcal{F}\) and its inverse \(\mathcal{F}^{-1}\).
\[f_{X_{(1)} - Y_{(m)}} = \mathcal{F}^{-1} \left\{ \mathcal{F} \left\{ n \left[ 1 - F_X(x) \right]^{n-1} f_X(x) \right\} \mathcal{F} \left\{ m f_Y(-y) \left[ F_Y(-y) \right]^{m-1} \right\} \right\}\]
Finally, we can obtain the cumulative distribution by integrating:
\[F_{X_{(1)} - Y_{(m)}}(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_{X_{(1)} - Y_{(m)}}(x,y) dx dy\]