Need an example of RV with a mean and no second moment

Question

An example like the t-distribution with 2 degrees of freedom would not suffice as the second moment exists but equals inf.

Answer 1

I think there is a confusion due to the fact that the distribution $t(1)$ ($t$ with 1 degree of freedom) has no mean, even if you accept values in $\overline{\mathbb R} = \mathbb R \cup \{ -\infty, +\infty \}$.

An indefinite integral like $$ E(X) = \int_{-\infty}^{+\infty} x f(x) \mathrm dx $$ is defined as the sum of the two following limits: $$ \int_0^{+\infty} x f(x) \mathrm dx = \lim_{T\rightarrow+\infty} \int_0^T x f(x) \mathrm dx, $$ $$ \int_{-\infty}^0 x f(x) \mathrm dx = \lim_{T\rightarrow-\infty} \int_T^0 x f(x) \mathrm dx.$$ If both limits are finite, it is well defined.

As $f(x)$ is a density, it is non-negative, and a $x f(x)$ has constant sign on $(-\infty,0]$ and $[0,\infty)$, hence if you accept values in $\overline{\mathbb R}$, it is possible to have infinite limits $ \int_0^{+\infty} x f(x) \mathrm dx = +\infty$ and $\int_{-\infty}^0 x f(x) \mathrm dx = -\infty$. If both are infinite then their sum is indefinite, even if you accept infinite values.

Concretely, on the following picture (graph of $xf(x)$ with $f$ the density of the $t$-distribution with one df) the positive ane negative areas are infinite, so their sum is indefinite.

However when dealing with the second moment $$ E\left( X^2 \right) = \int_{-\infty}^{+\infty} x^2 f(x) \mathrm dx,$$ as we have $x^2 f(x) \ge 0$ for all $x$, if the limits $$ \int_0^{+\infty} x^2 f(x) \mathrm dx = \lim_{T\rightarrow+\infty} \int_0^T x^2 f(x) \mathrm dx, $$ $$ \int_{-\infty}^0 x^2 f(x) \mathrm dx = \lim_{T\rightarrow-\infty} \int_T^0 x^2 f(x) \mathrm dx,$$ are not finite, they necessarily go to $+\infty$, hence either $E\left(X^2\right)$ is finite, or it is $+\infty$ if you accept values in $\overline{\mathbb R}$.

Concretely, the area under a curve which is always in the upper half plane is always defined in $\mathbb R \cup \{+\infty\}$. On the following picture (graph of $x^2 f(x)$ with $f$ the density of the $t$ distribution with 2 df), it is infinite.

Answer 2

The Holtsmark distribution ?