Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Skewness ▷ Partition into expected values

Theorem: Let $X$ be a random variable with expected value $\mu$ and standard deviation $\sigma$. Then, the skewness of $X$ can be computed as:

\[\label{eq:skew-partition} \mathrm{Skew}(X) = \frac{\mathrm{E}(X^3)-3\mu\sigma^2-\mu^3}{\sigma^3} \; .\]

Proof: The skewness of $X$ is defined as

\[\label{eq:skew} \mathrm{Skew}(X) = \frac{\mathrm{E}[(X-\mu)^3]}{\sigma^3} \; .\]

Because the expected value is a linear operator, we can rewrite \eqref{eq:skew} as

\[\label{eq:partition-s1} \begin{split} \mathrm{Skew}(X) &= \frac{\mathrm{E}[(X-\mu)^3]}{\sigma^3}\\ &= \frac{\mathrm{E}[X^3-3X^2\mu + 3X\mu^2 - \mu^3]}{\sigma^3}\\ &= \frac{\mathrm{E}(X^3) -3\mathrm{E}(X^2)\mu + 3\mathrm{E}(X)\mu^2 - \mu^3}{\sigma^3}\\ &= \frac{\mathrm{E}(X^3) -3\mu\left[\mathrm{E}(X^2)-\mathrm{E}(X)\mu\right]-\mu^3}{\sigma^3}\\ &= \frac{\mathrm{E}(X^3) -3\mu\left[\mathrm{E}(X^2)-\mathrm{E}(X)^2\right]-\mu^3}{\sigma^3} \; . \end{split}\]

Because the variance can be written in terms of expected values as

\[\label{eq:var-partition} \sigma^2 = \mathrm{E}(X^2)-\mathrm{E}(X)^2 \; ,\]

we can rewrite \eqref{eq:partition-s1} as

\[\label{eq:partition-s2} \mathrm{Skew}(X) = \frac{\mathrm{E}(X^3) -3\mu\sigma^2-\mu^3}{\sigma^3} \; .\]

This finishes the proof of \eqref{eq:skew-partition}.

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Metadata: ID: P407 | shortcut: skew-mean | author: tomfaulkenberry | date: 2023-04-20, 12:00.