Is there a class of probability density functions includes the laplacian and the normal pdfs?

Question

Is there a class of probability density functions that includes the Laplacian and the Normal pdfs?

$$f(x\mid\mu,b) = \frac{1}{2b} \exp \left( -\frac{|x-\mu|}{b} \right)$$ and $$f(x \mid \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2} } e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$$ seem very connected via some p for $p=1$ and $p=2$.

Is there a canonnical way to do this?

Maybe something like $$f_{\sigma, p} =\frac{1}{n(p)\sigma}\exp\left(-\left|\frac{x}{m(p)\sigma}\right|^p\right)?$$

I got the intuition that $n$ might be something along the lines of $$n(p) = \sqrt{2} p \Gamma(\frac{p+1}{p}),$$ but I can not figure out the appropriate $m$ and this is mainly speculation.

Answer 1

Yes, such a class is called generalized normal distribution. Please have a look here.

A PDF from this class has the form :

$$f(x)= \frac{\beta}{2\alpha\Gamma(1/\beta)} \; e^{-\left(\frac{|x-\mu|}{\alpha}\right)^\beta}$$

With :

• $\alpha$ and $\beta$ strictly positive real numbers,
• $\mu \in \mathbb{R}$,
• $\Gamma$ the Gamma function.

As you can see, for $\beta=1$, we have a Laplacian density function. And for $\beta=2$, we have the Normal one.