Division of random variablesPosted by Yuling Yao on Sep 18, 2019.
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.