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Definition

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

Conditional Probabilty

The probability for an event $A$ given that the event $B$ already occured is

$${\mathrm{P}}_B(A) = {\mathrm{P}}(A|B) = \frac{{\mathrm{P}}(A \cap B)}{{\mathrm{P}}(B)}$$

with events $A,B$.

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

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

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

Random Variables

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

Distribution

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

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