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Probability space

Last updated Dec 10, 2022

# 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:

# $\sigma$-algebra

The $\sigma$-algebra $A$ is a set of subsets of $\Omega$ with the properties

  1. $\Omega \in A$ and $\emptyset\in A$
  2. If $A_1 \in A$ and $A_2 \in A$ then also $A_1 \cup A_2$, $A_1 \cap A_2$ and $A_1 \\ A_2$
  3. 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

  1. $0 \leq \mu(A) \leq 1$
  2. Normalized: $\mu(\Omega) = 1$
  3. 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 $$\mu(A_1|A_2) = \mu(A_2|A_1)\frac{\mu(A_1)}{\mu(A_2)}$$