Probability distribution morphing from Gaussian to heavy tail

Question

I require a probability distribution which morphs from a something similar to a Gaussian (image 1) to something with a heavy tail on one side (image 2) based on some parameter. Can someone give me a hint what could be suitable?

There is a cut-off (here 64), and bigger values just get probability 0. The plots are not normalized, so don't bother about that.

Answer 1

You can always build you own by using a weighted sum of two distributions. $f=a f_1 + (1-a)f_2$ Careful on the choice of $f_2$, many of the likely candidates (e.g. $\chi^2$) are only defined for $x>0$. The truncation you will have to build in manually, but remember to renormalize afterwards.

Answer 2

You could try a mixture of two distributions, $Y(\beta)=(1-\beta)Z + \beta \cdot X$ where $Z$ is a normal distribution and $X$ some contaminating distribution (say a skewed heavy-tailed distribution). Now vary $\beta$ between $0$ and $1$ to get your ``morphing''.

Examples for $X$ could be stable distributions. The big disadvantage is that the densities are not known explicitly for most cases. One nice exception is the Lévy distribution. It has parameters which you can adjust to drive for example the mode.