If the joint probability density function of a vector of n random variables can be factored into a product of n functions of one variable f X 1 , … … , X n ( x 1 , … … , x n ) = f 1 ( x 1 ) ⋯ ⋯ f n ( x n ) , (where each f i is not necessarily a density) then the n variables in the set are all independent from each other, and the marginal probability density function of each of them is given by f X i ( x i ) = f i ( x i ) ∫ ∫ f i ( x ) d x .