# What is an example of a uni-modal symmetric non-Gaussian probability density function ?

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

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

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