The following image shows a problem and a solution to it. While I understand everything, I have a hard time understanding what E(X^4) < inf is for good? It doesn’t seem that essential and much of importance for this problem to be solved.
My first and only thought was that it tells us that we can be ensured that E(X^2) which is used throughout the problem is less than infinity,hence it can be evaluated to a numerical value since its not infinite?
Note: $V[X^2]=E[X^4]-E[X^2]^2$. The condition $E[X^4]<\infty$ ensures that this variance is well-defined.