  This paper mainly addresses the optimization of $p$-th moment of
$\mathbb{R}^n$-valued random variable. Through an ingenious approximation
mechanism, one transforms the maximization problem into a sequence of
minimization problems, which can be converted into a sequence of nonlinear
differential equations with constraints by variational approach. The existence
and uniqueness of the solution for each equation can be demonstrated by
applying the canonical duality method. Moreover, the dual transformation gives
a sequence of perfect dual maximization problems. In the final analysis, one
constructs the approximation of the probability density function accordingly.
