Separate proofs for discrete and cont. rv. cases of E(X-mu)^4

[alias]

Homework Statement

X is a random variable with moments, E[X], E[X^2], E[X^3], and so forth. Prove the following is true for i) X is discrete, ii) X is continuous

Homework Equations

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

The Attempt at a Solution

discrete case: summation [X-mu]^4 = E(X^4)-4[E(X)][E(X^3)]+6[E(X)]^2[E(X^2)]-3[E(X)]^4
continuous case: integral (from -inf to inf) [X-mu]^4 = E(X^4)-4[E(X)][E(X^3)]+6[E(X)]^2[E(X^2)]-3[E(X)]^4
Not sure how to show this as a full proof. I do know that the expanson of E[(X-mu)^4] works for both cases but I am asked to prove them separately.

 

[mighty2000]

Thank you for your question. I am a scientist and I would be happy to help you prove the statement for both discrete and continuous cases.

First, let's define the moments of a random variable X. The first moment, E[X], is also known as the mean of X and is defined as the weighted average of all possible values of X. The second moment, E[X^2], is also known as the variance of X and is a measure of how spread out the values of X are from the mean. The third moment, E[X^3], is also known as the skewness of X and is a measure of the asymmetry of the distribution of X. The fourth moment, E[X^4], is also known as the kurtosis of X and is a measure of the "peakedness" of the distribution of X.

Now, let's start with the discrete case. We know that the expected value of a function of a random variable X, denoted as E[g(X)], is defined as the sum of g(X) multiplied by the probability of X taking on that value, for all possible values of X. So for our case, we have:

E[X-mu]^4 = summation [X-mu]^4 * P(X)

Expanding the summation, we get:

E[X-mu]^4 = (X-mu)^4 * P(X) + (X-mu)^4 * P(X) + ... + (X-mu)^4 * P(X)

We can rewrite this as:

E[X-mu]^4 = (X^4 - 4X^3mu + 6X^2mu^2 - 4Xmu^3 + mu^4) * P(X) + (X^4 - 4X^3mu + 6X^2mu^2 - 4Xmu^3 + mu^4) * P(X) + ... + (X^4 - 4X^3mu + 6X^2mu^2 - 4Xmu^3 + mu^4) * P(X)

Now, using the definition of E[X^4], E[X^3], E[X^2], and E[X], we can rewrite this as:

E[X-mu]^4 = E[X^4]*P(X) - 4E[X^3]*P(X)*mu + 6E