Distribution where the variance diverges for some parameter values but not others I'm wondering whether there's a standard / nice / tractable family of distributions where the variance is defined for some parameter values but not others, while the mean stays finite.
I'm imagining a parameter that controls the variance, such that there's a critical value at which the variance goes to infinity and is undefined above the critical value.
It would be ideal if the support was the non-negative reals for all parameter values, with the pdf decreasing monotonically even in the case where the variance is undefined.

A simple example would be the T-distribution with degrees-of-freedom no less than two.  If you want the mean to be non-trivial then you could use a shifted T-distribution with a mean parameter.  If you want you could take the family of shifted T-distributions with degrees-of-freedom no less than two and then reparameterise it with a mean parameter $\mu$ and variance parameter $\sigma^2$.  This gives the density function:
$$\text{St}(x | \mu, \sigma^2)
= \sqrt{\frac{\sigma^2-1}{2 \sigma^2 \pi}} \cdot \frac{\Gamma \big( \tfrac{\sigma^2}{\sigma^2-1} + \tfrac{1}{2} \big)}{\Gamma \big( \tfrac{\sigma^2}{\sigma^2-1} \big)} \cdot \Bigg( 1 + \frac{\sigma^2-1}{2\sigma^2} \cdot (x-\mu)^2 \Bigg)^{- \tfrac{\sigma^2}{\sigma^2-1} - \tfrac{1}{2}},$$
where $\mu \in \mathbb{R}$ is the mean parameter and $\sigma^2 \in [1, \infty]$ is the variance parameter.
