A while ago, I was told that if you roll a d%, by rolling 2 ten sided dice and treating one as the tens digit and the other as the ones, then you can gain 2 independently distributed values by flipping which is 10s digit and which is ones.
I want to prove or disprove this, It feels like it shouldn't be true.
Ie: for X, Y independent and Uniformly discretely distributed, on 0,1,2..,9 Show, that for random variables A=10X+Y B=10Y+X A and B are Independent (or show the converse)
So it is clear that:
$P_X (x) = P_Y (y)=1/10$
$P_XY (x,y) = 1/100$ (from independence)
$P_A (a) = P_B(b) = 1/100$
Now I must show that $P_AB(a,b)=1/10000=P_A(a)\times P_B(b)$ to show independence. Just not to sure where to start with this, I tried to use the convolution rule for sums of random variables, but I must have made a mistake since i ened up with $P_AB(a,b)=1$