Someone just asked a question about rank of a vector. Any non-zero vector should have rank $1$ given the (only) definition I am aware of.
But I started wondering what would happen if we don't have a vector of real or complex numbers, but instead a vector of probability distributions.
Could we define some probabilistic rank concept for it somehow?
Example:
$$v = \alpha[1,0]^T + \beta[0,1]^T$$ $$\cases{ \alpha \in U(0,1)\\ \beta\in U(0,\epsilon)}$$
If $\epsilon = 0$ ($\beta = 0$ with $100\%$ chance), then $v$ is clearly always a multiple of one vector and shall have rank 1.
But what if $\epsilon > 0$. How could we define some probabilistic rank concept then?
$v$ is a vector of $\mathbb{R}^{1 \times 2}$ regardless of the value of $\epsilon$.
If the $\alpha=\beta=0$, then $v=0$, the rank is $0$, however the probability of such event is $0$.
If $\alpha$ or $\beta$ are non-zero, then the rank is $1$.
The probability of having it to be rank $1$ is $1$.
In general, we can discuss what is the probability that a matrix take a particular rank and also compute quantity such as the expected value.