Division of random variables

Posted by Yuling Yao on Sep 18, 2019.       Tag: casual  

This is trivial math question but it once bothers me in the construction of Dirichlet process using stick breaking process. The context is that if $V \sim beta(1, M)$ and $\theta \sim Dirichlet (\bar \alpha)$ then

Then P is a Dirichlet precess $DP(M \bar \alpha)$

Every neuron inside my brain says “do the division!”

$V$ is a scaler-rm independent of $\theta$, which seems to imply

It is just wrong in every aspect.

The point here is that the division of random variables can be messy without careful treatment. e.g, if X is Bernoulli (0.5), and Y is an independent Bernoulli (0.5), then $Z=X\times Y = Bernoulli (0.25)$. But what is Bernoulli (0.25)/ Bernoulli (0.5)? Both the dependence and 0 makes the division ambiguous.