= Solution
The <expected value> is not defined: the integral of $yf(y)$ over the positive half-line diverges logarithmically to $+\infty$, and the negative half-line gives $-\infty$. In particular, symmetry alone does not supply a mean. More explicitly,
$$
\int_0^R yf(y)\,dy=\frac{x_0}{2\pi}\ln\left(1+\frac{R^2}{x_0^2}\right)\longrightarrow\infty.
$$
The second <moment> also diverges because $y^2f(y)\to x_0/\pi>0$ as $|y|\to\infty$. Thus
$$
\boxed{\text{the mean, variance and standard deviation are undefined}.}
$$
The symmetric <Cauchy principal value> of the first-moment integral is zero, but it is not an <expected value> and does not define a <variance>.
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