Distribution functions of independent minima and maxima ($F_{\min }=1-(1-F_X)(1-F_Y),F_{\max }=F_X{F}_Y$)

Independence factors the events that both variables exceed a threshold or both lie below it. This gives the two cumulative distribution functions above, including for distributions with atoms. The minimum formula is most naturally expressed through survival probabilities, while the maximum formula uses the CDFs directly.