# Skewness of a sum with a positive summand

Let $X$ and $Z$ be two random variables with finite third moment, and let $Z>0$. Is it true that the skewness of $X+Z$ is greater or equal than that of $X$? Such a relation clearly holds for the mean while it does not for the variance. How about the skewness and the other odd moments?

I fiddled about with matlab and found no counterexamples by assuming $X$ normal and $Z$, possibly correlated with $X$, to be log-normal, chi-squared ecc.. can anyone help?

• I think I have found my own answer. For a counterexample it suffices to find a positive and negatively-skewed random variable $Z$; for if $X$ is normal and independent of $Z$ then $Skew(X+Z)=Skew(Z)<0=Skew(X)$. Such $Z$ is for example, a Weibull distribution of parameters $\lambda=0.2$ and $k=1$. Apr 19, 2013 at 8:45

Let $X$ be normally distributed and $Z$ be independent of $X$, $X$ having a Weibull distribution of parameters $\lambda=0.2$ and $\kappa=1$, so that $Skew(Z)<0$. Then $Skew(X+Z)=Skew(Z)<0=Skew(X)$.