Probability space
# Definition probability space
A probability space is the formal description of a random process
Definition: Probability space A probability space is a triple $(\Omega, \mathcal{F}, P)$ with
- $\Omega$: Sample space: Set of all possible outcomes
- $\mathcal{F}$: Event space: Arbitrary set of events - an event is a set of outcomes
- $P$: Probability function: Assigns each event in the event space a probability
More formally we have:
- $\Omega$ is a set
- $\mathcal{F}$ is a $\sigma$-algebra of events
- $P$ is a probability measure on the $\sigma$-algebra
# $\sigma$-algebra
The $\sigma$-algebra $A$ is a set of subsets of $\Omega$ with the properties
- $\Omega \in A$ and $\emptyset\in A$
- If $A_1 \in A$ and $A_2 \in A$ then also $A_1 \cup A_2$, $A_1 \cap A_2$ and $A_1 \\ A_2$
- The union of a selection of events in $A$ is also in $A$
These definitions basically ensure you can logically connect events
# Probability measure
Definition: Probability measure A map $\mu : A \rightarrow \Reals$ which assigns each event a real number interpreded as probability
- $0 \leq \mu(A) \leq 1$
- Normalized: $\mu(\Omega) = 1$
- Union of events has probability of sum of events: $\mu(\cup A_n) = \sum_n \mu(A_N)$
From this one can further derive probability theory.
# Conditional probabilities
Definition: Conditional probability Probability of an event $A_1$ under the condition that $A_2$ happend. $$\mu(A_1|A_2) = \frac{\mu(A_1\cap A_2)}{\mu(A_2)}$$
Definition: Statistical independence Two events $A_1$ and $A_2$ are statistically independent if $$\mu(A_1 | A_2) = \mu(A_1)$$ or equivilantly $$\mu(A_1 \cap A_2) = \mu(A_1)\cdot\mu(A_2)$$ which can be extendent to arbitrary many events that have to factorize
From the definition of conditional probability immediatly follows Bayes’s theorem.
Definition: Bayes theorem For two events $A_1$ and $A_2$ from a probabiliy space we have $$\mu(A_1|A_2) = \mu(A_2|A_1)\frac{\mu(A_1)}{\mu(A_2)}$$