I just started studying discrete probability distributions and after a little introduction there are a couple of rules (axioms most probably) about (that's what my notes state but I'm not a hundred percent sure) probability measures.
Here are the points:
- $\forall i \in [1,n]\cap \mathbb{N},\thinspace\thinspace\thinspace\thinspace\omega_j\in\Omega \quad \mathbb{P}( \{ w_j \}_{j=1}^{\infty}) = P_i$ and $\forall E\subseteq\Omega \quad \mathbb{P}(E) = \sum\limits_{j} \mathbb{P}(\omega_j)$
- $ P_i \geq 0 \quad \forall \text{ admissible } i $
- $\sum\limits_{i=1}^n P_i = 1$
I'm having hard times understanding what the first one is trying to say, (suspect some kind of badly formalized additivity?) and actually have some suspect that I copied them wrong...
Does anybody know what the first one should actually be?
I wrote my teacher, who told me that the correct one was:
$\text{Let} \;\; \mathbb{P}(\{\omega_j\}_{j=1}^n) = P_j \;\;\;\forall \omega_j\in\Omega \;\; $ where $\{\omega_j\}$ has at most countably many elements.
Then $\forall$ Event $E \subset \Omega\quad \mathbb{P}(E) = \sum\limits_{j| \omega_j\in E} P_j$