# Is there a probability distribution with the following properties?

I'm looking for a univariate probability distribution defined for $x \in (-\infty, \infty)$ with the following properties:

• The PDF is symmetric around the origin ($p(x)=p(-x)$).
• The derivative of the PDF at $x=0$ (with the limit taken from above) can be specified to lie in some interval $-\infty \dots 0$ (not necessarily inclusive). Note that this implies that the derivative of the PDF may be discontinuous at that point.
• The distribution should interpolate between a heavy tailed and a normal distribution. The normal distribution must be a special case.
• The CDF and inverse CDF can be efficiently evaluated (i.e., using well-known functions and without integrating numerically in a typical computer algebra system).
• The PDF should be as smooth as possible (everywhere except at the origin).

The generalized normal distribution (with $p(x)\sim\exp(-|x|^\beta)$) comes close to this, but $\beta$ specifies the tail behavior as well as its “peakiness” at $x=0$. I would like to be able to specify these two things as independently as possible.

One idea would be to convolve the PDF of a generalized normal with a normal PDF of a specified scale (i.e., adding a normal random variable to the generalized normal), thereby “smoothing” it at the origin. However, I am stuck with computing the resulting distribution in closed form.