Skewness of Moment Generating Functions

[rafsan7238]

Let X and Y be two independent random variables with the moment generating functions MX(t) = 1 − t + αt^2 + e1(t) (α > 0.5) and MY(t) = 1 + t + t^2 + e2(t), where the functions e1(t), e2(t) and their first three derivatives converge to zero as t → 0. Define Z = X + Y . Find the maximum value of skewness of Z. Find the variance Var(Z) of Z with maximum skewness.

 

[BigBeachBanana]

[math] M_Z(t) = M_X(t) * M_Y(t)\\ E[Z]= M_Z'(0)\\ Var(Z) = M_Z''(0) + (M_Z'(0))^2\\ E[(Z - E[Z])^3] = E[Z^3] - 3E[Z^2]E[Z] + 2E[Z]^3\\ \text{Skewness} = \dfrac{E[Z^3] - 3E[Z^2]E[Z] + 2E[Z]^3}{\sigma^3} [/math]
Compute the skewness first, then use Calculus to maximize it.