What are examples of Unimodal Symmetric probability density functions that are not Gaussian ?

I searched online and found this article: https://en.wikipedia.org/wiki/Unimodality

This article gives 4 examples:

  1. Cauchy distribution

  2. Student's t-distribution

  3. chi-squared distribution
  4. exponential distribution

But chi-squared and exponential are not symmetric. The student-t distribution looks exactly like a Gaussian.

The only one that seems to be non Gaussian and symmetric is Cauchy. But the this article says it is a ratio of two normal random variables.

So I'm wondering if there are any unimodal symmetric distributions that have nothing to do with Gaussian (or normal) distributions?

  • How do you define "have nothing to do with"? – Robert Israel Dec 8 '17 at 1:46
up vote 0 down vote accepted

There is a symmetric version of the exponential distribution called the Laplace distribution, with pdf:

$f(x)=\frac{1}{2a} e^{-| x | /a}, -\infty < x < \infty$

The distribution of the difference of two uniform distributions is an example. The double exponential distribution, with density $\exp(-|x|)/2$. The convolution of the above two.

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