Rewriting the fourth moment in terms of

[alias]

rewriting the fourth moment in terms of...

Homework Statement

X is a random variable with moments E[X], E[X^2], E[X^3]. Suppose var(X)= σ^2, and that skewness is given as, S=[E(X-μ)^3)]/(σ^3) Rewrite the right hand side of the expression(see below) in terms of μ, σ^2, S and μ4

Homework Equations

E[X-μ]^4 = E(X^4) - 4[E(X)][E(X^3)] + 6[E(X)]^2[E(X^2)] - 3[E(X)]^4

The Attempt at a Solution

E[(X^4)] = μ, that's all I've got.
I've been coming back to this all day and I can't see where σ^2, S and μ4 fit into the right side of the equation. Can anyone please give me an idea? Thanks.

 

[mighty2000]

Dear fellow scientist,

Thank you for your post. The right hand side of the expression can be rewritten as follows:

E[(X-μ)^4] = E[X^4 - 4X^3μ + 6X^2μ^2 - 4Xμ^3 + μ^4] (using the binomial expansion of (X-μ)^4)

= E[X^4] - 4E[X^3]μ + 6E[X^2]μ^2 - 4E[X]μ^3 + μ^4 (using linearity of expectation)

= E[X^4] - 4[E[X]][E[X^3]] + 6[E[X]^2][E[X^2]] - 4[E[X]^3] + [E[X]^4] (using properties of expectation and the given moments)

= E[X^4] - 4[E[X]][E[X^3]] + 6[E[X]^2][E[X^2]] - 3[E[X]^4] + 2[E[X]^4] (adding and subtracting [E[X]^4])

= E[X^4] - 4[E[X]][E[X^3]] + 6[E[X]^2][E[X^2]] - 3[E[X]^4] + 2μ^4 (since E[X]^4 = μ^4)

= E[X^4] - 4[E[X]][E[X^3]] + 6[E[X]^2][E[X^2]] - 3[E[X]^4] + 2μ^4 (using the given definition of skewness)

= E[X^4] - 4[E[X]][E[X^3]] + 6[E[X]^2][E[X^2]] - 3σ^4 + 2μ^4 (since var(X) = σ^2)

Therefore, the right hand side of the expression can be rewritten as E[(X-μ)^4] = E[X^4] - 4[E[X]][E[X^3]] + 6[E[X]^2][E[X^2]] - 3σ^4 + 2μ^4. This expression involves μ, σ^2, S, and μ^4, as desired.

I hope this helps clarify the solution for you. Let me know