Probability MT22, Axioms of probability space
Flashcards
Sample spaces and events
In probability, what does $\Omega$ represent?
The set of all possible outcomes, i.e. the sample space.
If the observed outcome of an experiment is $\omega$, $\omega \in \Omega$, @define what it means for event $A$ to occur.
In the case where $\Omega$ is finite and all outcomes are equally likely, what’s the formula for the probability of $A$?
What’s the formal definition of a probability space?
A triplet $(\Omega, \mathcal{F}, \mathbb{P})$.
What two sets does the function $\mathbb{P}$ map between?
What’s a common trick involving proofs about probabilities?
Writing $\mathbb{P}(A) = \mathbb{P}(A \cap \Omega)$
Axioms of a probability space
What’s the axiom relevant to $\mathcal{F}$ in probability about the overall sample space, $\Omega$?
What’s the axiom relevant to $\mathcal{F}$ in probability about an event $A$ and its complement $A^C$?
What’s the axiom relevant to $\mathcal{F}$ in probability about the union of events?
What’s the axiom relevant to $\mathbb{P}$ in probability about non-negativity?
What’s the axiom relevant to $\mathbb{P}$ in probability about the sample space $\Omega$?
What’s the axiom relevant to $\mathbb{P}$ in probability about disjoint events $A$ and $B$?
@State the three axioms related to $\mathcal F$ in a probability space $(\Omega, \mathcal F, \mathbb P)$.
- $\Omega \in \mathcal F$,
- If $A \in \mathcal F$, then $A^C \in \mathcal F$,
- If $\{A _ i : i \in I\}$ is a finite or countably infinite collection of members of $\mathcal F$, then $\cup _ {i \in I} A _ i \in \mathcal F$.
@State the three axioms related to $\mathbb P$ in a probability space $(\Omega, \mathcal F, \mathbb P)$.
- $\mathbb P(A) \ge 0$ for all $A \in \mathcal F$.
- $\mathbb P (\Omega) = 1$
- If $\{A _ i : i \in I\}$ is a finite or countably infinite collection of members of $\mathcal F$, and they are pairwise disjoint, then $\mathbb P\left(\cup _ {i \in I} A _ i\right) = \sum _ {i \in I} \mathbb P (A _ i)$.