There is something that might be fun in mathematics, which is the connexion between trivial, tautological and difficult questions. Sometimes, things are so intuitive, that they seem to be obvious. But mathematicians aren’t jedis, and they should not trust too much their intuition… I mean intuition is fine, but it is not a proof. It is like those standard results we learn in topology courses, e.g. “the closure of an open ball is not necessarily the closed ball”. The other thing is that after a while, you try to prove something, until someone makes you realize that it is the definition…
has a priori distribution
, and that
, given
, has (conditional) distribution
.
, is- step 1: generate

- step 2: given that generation of
, generate 
? Or is it just trivial ?”. After having those previous philosophical questions, we came to the point that if it was trivial, then we should be able to prove it. A standard way of writing the algorithm is to use the quantile based technique-
with
, -
with
,
n=1 theta=rgama(n,3,3) X=rpois(n,lambda=theta)
where
and
are two independent random variables with a uniform distribution on the unit interval. Let us try to derive its distribution, i.e.



.OpenEdition suggests that you cite this post as follows:
Arthur Charpentier (April 27, 2012). Proving tautological versus trivial results in mathematics. Freakonometrics. Retrieved January 20, 2026 from https://doi.org/10.58079/oulb
