A family of distributions induced by a metric

Asked Nov 09 '19 at 19:42

Active Nov 09 '19 at 19:42

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I'm interested in the family of distribution which can be expressed in the following form

$f(x|\mu)=C \exp(-d(x,\mu))$

where $\mu$ is a parameter of the distribution and $d(*,*)$ is a metric. Normal distribution and Laplace distribution satisfy this property. I was wondering if this family has already been studied and what is its name.

asked Nov 09 '19 at 19:42

armando

Maybe a more general family is represented by:
$$f(x|\mu,\alpha)=C \exp(-\alpha \cdot d(x,\mu))$$
– the_candyman Nov 09 '19 at 19:56
@the_candyman - assuming $\alpha \gt 0$ then wouldn't $\alpha\cdot d(x,\mu)$ be another metric? – Henry Nov 09 '19 at 21:09
@Henry, that's right! – the_candyman Nov 10 '19 at 09:28
I was wrong, normal distribution is not in this family. – armando Nov 11 '19 at 16:47

A family of distributions induced by a metric

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