Finding upper and lower bounds for fourth moment using a linear program

Question

The problem is from Bertsimas' book "Introduction to linear optimization". I would just like a hint or if someone could give me some direction of where to go with this problem.

Suppose that Z is a random variable taking values in the set $0,1,...,K$ with probabilities $p_0,p_1,...,p_K$, respectively. We are given the values of the first two moments $E[Z] = \sum_{k = 0}^{K}kp_k$ and $E[Z^2] = \sum_{k = 0}^{K}k^2p_k$ of Z, and we would like to obtain upper and lower bounds on the value of the fouth moment $E[Z^4] = \sum_{k=0}^K k^4p_k$ of Z. Show how linear programming can be used to approach this problem.

Thanks for any help or advice in this problem.

Answer 1

Linear programming can be used to determine the probabilities. Minimize or maximize $\sum_{k=0}^K k^4 p_k$ such that the probabilities are nonnegative and sum to $1$.