= Solution
{c}
Let $X$ have the <Pareto distribution> with <probability density function> $f_X(x)=3x^{-4}$ for $x\geq1$, and zero otherwise. Its <normalization of a probability distribution> and first two <moments> follow from
$$
\int_1^\infty3x^{-4}\,dx=1,\qquad
\mathbb EX=3\int_1^\infty x^{-3}\,dx=\frac32,\qquad
\mathbb EX^2=3\int_1^\infty x^{-2}\,dx=3.
$$
Set $Y=X-3/2$. Then
$$
\boxed{\mathbb EY=0,\qquad\operatorname{Var}(Y)=3-\frac94=\frac34.}
$$
The <probability distribution> of $Y$ is not a <symmetric distribution>: $Y\geq-1/2$ <almost surely>, whereas $\mathbb P(Y>1/2)=\mathbb P(X>2)=1/8>0$ and $\mathbb P(Y<-1/2)=0$. For $X\geq3$ we have $|Y|\geq X/2$, so
$$
\mathbb E|Y|^3\geq\frac18\mathbb E\bigl[X^3\mathbf1_{\{X\geq3\}}\bigr]
=\frac38\int_3^\infty\frac{dx}{x}=\infty.
$$
Thus $\boxed{Y=X-3/2}$ has every property required.
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