A probabilistic method to prove AM-QM inequality

Inequality is one of the interesting topics in mathematics that attracts the attention of both students and researchers. Solving inequalities often requires creativity, making it a challenging topic for many students. The inequality of arithmetic and quadratic means, or AM-QM in short, states that the arithmetic mean of a list of non-negative real numbers is less than or equals to the square root of the quadratic mean of the same list. There are many methods of proving the AM-QM inequality. In this paper, we will present a probabilistic method to prove the weighted general AM-QM inequality and show that the classical AM-QM inequality is a special case of the generalized AM-QM inequality with equal weights.