/ math / statistics /

[edit]

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 Ξ©\Omega nonempty set of outputs of experiment
Sigma Algebra π”½βŠ†2Ξ©\mathbb F \subseteq 2^\Omega set of subsets of outputs (events)
Probability P:𝔽↦[0,1]\mathop{P}: \mathbb F \mapsto [0,1]
Random Variable 𝑋:Ω↦𝕏\textit{X}: \Omega \mapsto \mathbb X mapped subsets of Ξ©\Omega
Observations: x1,…,xNx_1, \ldots, x_N single values of 𝑋\textit{X}
Observation Space 𝕏\mathbb X possible observations of 𝑋\textit{X}
Unknown parameter θ∈Θ\theta \in \Theta parameter of propability function
Estimator 𝑇:π•β†¦Ξ˜\textit{T}: \mathbb X \mapsto \Theta 𝑇(𝑋)=ΞΈΜ‚\textit{T}(\textit{X}) = \hat{\theta}, finds ΞΈΜ‚\hat{\theta} from 𝑋\textit{X}

Conditional Probabilty

The probability for an event AA given that the event BB already occured is

PB(A)=P(A|B)=P(A∩B)P(B)\mathop{P}_B(A) = \mathop{P}(A|B) = \frac{\mathop{P}(A \cap B)}{\mathop{P}(B)}

with events A,BA,B.

Multiplication: P(A∩B)=P(A|B)P(B)=P(B|A)P(A)\mathop{P}(A \cap B) = \mathop{P}(A|B)\mathop{P}(B) = \mathop{P}(B|A)\mathop{P}(A)

Bayes’ law: P(Bk|A)=P(A|Bk)P(Bk)βˆ‘i∈IP(A|Bi)P(Bi)\mathop{P}(B_k | A) = \frac{\mathop{P}(A | B_k)\mathop{P}(B_k)}{\sum\limits_{i \in I} \mathop{P}(A|B_i) \mathop{P}(B_i)}

Total probabilty: P(A)=βˆ‘i∈IP(A|Bi)P(Bi)\mathop{P}(A) = \sum\limits_{i \in I} \mathop{P}(A|B_i)\mathop{P}(B_i)

Random Variables

𝑋:Ω↦Ω′\textit{X}: \Omega \mapsto \Omega' is a random variable if for all events Aβ€²βˆˆπ”½β€²A' \in \mathbb F' there exists an event Aβˆˆπ”½A \in \mathbb F such that {Ο‰βˆˆΞ©|𝑋(Ο‰)∈Aβ€²}βˆˆπ”½\left\{\omega \in \Omega|\textit{X}(\omega) \in A'\right\} \in \mathbb F.

Distribution

PDF: f𝑋(x)=dF𝑋(x)dxf_{\textit{X}}(x) = \frac{\,\text{d}F_{\textit{X}}(x)}{\,\text{d}x} CDF: F𝑋(x)=βˆ«βˆ’βˆžxf𝑋(ΞΎ)dΞΎF_{\textit{X}}(x) = \int\limits_{-\infty}^{x}{f_{\textit{X}}(\xi)\,\text{d}\xi}