It is well known (see [2], p. 158) that if X and Y are independent random variables with a continuous joint probability density function (pdf) which is spherically symmetric about the origin, then both X and Y are normally distributed. In this note we examine the condition that the joint pdf be spherically symmetric about the origin and show that the normal distribution is strongly dependent on the choice of metric for R2.
@article{urn:eudml:doc:38835,
title = {Generalized normal distributions.},
journal = {Stochastica},
volume = {4},
year = {1980},
pages = {221-225},
zbl = {0457.60018},
mrnumber = {MR0611505},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:38835}
}
Tardiff, Robert M. Generalized normal distributions.. Stochastica, Tome 4 (1980) pp. 221-225. http://gdmltest.u-ga.fr/item/urn:eudml:doc:38835/