# Probability Theory#

Probabilities denote the chance that an event occurs, it is not any kind of guarantee that this event will happen after a certain amount of trials.

## Terms#

Name Definition
Sample Space $$Ω$$ nonempty set of outputs of experiment
Sigma Algebra $$\mathbb F \subseteq 2^Ω$$ set of subsets of outputs (events)
Probability $$\P: \mathbb F \mapsto [0,1]$$
Random Variable $$\X: Ω \mapsto \mathbb X$$ mapped subsets of $$Ω$$
Observations: $$x_1, \ldots, x_N$$ single values of $$\X$$
Observation Space $$\mathbb X$$ possible observations of $$\X$$
Unknown parameter $$θ ∈ Θ$$ parameter of propability function
Estimator $$\T: \mathbb X \mapsto Θ$$ $$\T(\X) = \hat{θ}$$, finds $$\hat{θ}$$ from $$\X$$

## Conditional Probabilty#

The probability for an event $$A$$ given that the event $$B$$ already occured is $$\(\P_B(A) = \P(A|B) = \frac{\P(A \cap B)}{\P(B)}$$\) with events $$A,B$$.

Multiplication: $$\P(A \cap B) = \P(A|B)\P(B) = \P(B|A)\P(A)$$

Bayes' law: $$\(\P(B_k | A) = \frac{\P(A | B_k)\P(B_k)}{\sum\limits_{i \in I} \P(A|B_i) \P(B_i)}$$\)

Total probabilty: $$\P(A) = \sum\limits_{i \in I} \P(A|B_i)\P(B_i)$$

## Random Variables#

$$\X : \Omega \mapsto \Omega'$$ is a random variable if for all events $$A' \in \mathbb F'$$ there exists an event $$A \in \mathbb F$$ such that $$\left\{\omega \in \Omega|\X(\omega) \in A'\right\} \in \mathbb F$$.

## Distribution#

PDF: $$f_{\X}(x) = \frac{\diff F_{\X}(x)}{\diff x}$$ CDF: $$F_{\X}(x) = \int\limits_{-\infty}^{x}{f_{\X}(\xi)\diff\xi}$$