The distribution F ( x ) = ∏ i = 1 n F i ( x ) {\displaystyle F(x)=\prod _{i=1}^{n}F_{i}(x)} can be generated from F ( x ) = P ( m a x ( F 1 , . . . , F n ) ≤ k ) {\displaystyle F(x)=P(max(F_{1},...,F_{n})\leq k)} because the product of independent RV's are the same as the max of the RV's. If the max probability of all of the F i {\displaystyle F_{i}} 's is less than k, then all of the individual F i {\displaystyle F_{i}} 's must be less than k.