Question

If Y=aX+b, show that Y has the same coefficient of skewness?

Answer 1

The skewness of a random variable X is the third standardized moment, defined as $\tilde{\mu}^3(X)=\sigma(X)^{-3}E[(X-\mu(X))^3]$ , where $\mu(X),\ \sigma(X)$ are the mean and the standard deviation of X. If $Y=aX+b$, then

$\mu(Y)=E(aX+b)=a\mu(X)+b$

$Var(Y)=E((Y-\mu(Y))^2)=E((aX+b-(a\mu(X)+b))^2)=a^2E((X-\mu(X))^2)=a^2Var(X)$

$\sigma(Y)=\sqrt{Var(Y)}=\sqrt{a^2Var(X)}=|a|\sigma(X)$

$\tilde{\mu}^3(Y)=\sigma(Y)^{-3}E[(Y-\mu(Y))^3]=|a|^{-3}\sigma(X)^{-3}E[(aX+b-(a\mu(X)+b))^3]=\frac{a^3}{|a|^3}E((X-\mu(X))^3)=\mathrm{sign}(a) \tilde{\mu}^3(X)$

Therefore, if a>0, then Y has the same coefficient of skewness as X. If a<0 then X, Y have the same absolute values of the coefficient of skewness, but their signs are different.

If a=0, then the coefficient of skewness of Y is undefined.