Abstract:

For two-dimensional continuous random variables（X,Y）, two independent necessary and sufficient conditionsare obtained
based on the joint density function and the joint distribution function, and the application is also given. At last, the results are
generalized to（X1, X2, ..., Xn） case of multi- dimensional random variables, and two independence of necessary and sufficient
conditions for determining（X1, X2, ..., Xn）independence are given. The results improved the original way and made continuous random
variables independent judgment easier.

Key words: multi-dimensional random variables, independence, necessary and sufficient condition