## hands-on probability 101

**W**hen solving a rather simple probability question on X validated, namely the joint uniformity of the pair

when A,B,C are iid U(0,1), I chose a rather pedestrian way and derived the joint distribution of (A-B,C-B), which turns to be made of 8 components over the (-1,1)² domain. And to conclude at the uniformity of the above, I added a hand-made picture to explain why the coverage by (X,Y) of any (red) square within (0,1)² was uniform by virtue of the symmetry between the coverage by (A-B,C-B) of four copies of the (red) square, using color tabs that were sitting on my desk..! It did not seem to convince the originator of the question, who kept answering with more questions—or worse an ever-changing question, reproduced in real time on math.stackexchange!, revealing there that said originator was tutoring an undergrad student!—but this was a light moment in a dreary final day before a new lockdown.

April 3, 2021 at 10:15 am

Ces centaines de lignes pour montrer que (X,Y)-(A-B,C-B) modulo 1 est uniforme dans le carre, alors que $\int_0^1 exp(2i \pi nx)dx=0$ ou 1 suivant $n\neq 0 $ ou $n=0$ suffit a traiter le probleme par le calcul de E(exp (2i \pi(nX+mY)))……

April 3, 2021 at 11:51 am

Je concède !

April 4, 2021 at 10:05 am

Enfoncons le clou: soit 3 va A,B,C independantes de [0,1] telles que

deux d’entre elles soient uniformes.

Alors $X=A_B, Y=C-B$ sont independantes et uniformes modulo 1