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Was previously P(A \cup B) or P(A \cup B) which is same thing. Should be P(A \cup B) or P(B \cup A) |
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==Probability theory==
* [[Random variable]]s are usually written in [[upper case]] roman letters: ''X'', ''Y'', ''Z'', ''T'', etc.<ref name=":0">{{Cite web|date=2020-04-26|title=List of Probability and Statistics Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/probability-statistics-symbols/|access-date=2020-09-10|website=Math Vault|language=en-US}}</ref>
* Particular realizations of a random variable are written in corresponding [[lower case]] letters. For example, ''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x''<sub>''n''</sub> could be a [[random sample|sample]] corresponding to the random variable ''X''. A cumulative probability is formally written <math>P(X\le x) </math> to differentiate the random variable from its realization.
* The probability is sometimes written <math>\mathbb{P} </math> to distinguish it from other functions and measure ''P'' so as to avoid having to define “''P'' is a probability” and <math>\mathbb{P}(X\in A) </math> is short for <math>P(\{\omega \in\Omega: X(\omega) \in A\})</math>, where <math>\Omega</math> is the event space and <math>X(\omega)</math> is a random variable. <math>\Pr(A)</math> notation is used alternatively.<ref name=":0" />
*<math>\mathbb{P}(A \cap B)</math> or <math>\mathbb{P}[B \cap A]</math> indicates the probability that events ''A'' and ''B'' both occur. The [[joint probability distribution]] of random variables ''X'' and ''Y'' is denoted as <math>P(X, Y)</math>, while joint probability mass function or probability density function as <math>f(x, y)</math> and joint cumulative distribution function as <math>F(x, y)</math>.<ref name=":0" />
*<math>\mathbb{P}(A \cup B)</math> or <math>\mathbb{P}[B \cup A]</math> indicates the probability of either event ''A'' or event ''B'' occurring (“or” in this case means [[inclusive or|one or the other or both]]).<ref name=":0" />
*[[sigma-algebra|σ-algebras]] are usually written with uppercase [[Calligraphy|calligraphic]] (e.g. <math>\mathcal F</math> for the set of sets on which we define the probability ''P'')
*[[Probability density function]]s (pdfs) and [[probability mass function]]s are denoted by lowercase letters, e.g. <math>f(x)</math>, or <math>f_X(x)</math>.<ref name=":0" />
*[[Cumulative distribution function]]s (cdfs) are denoted by uppercase letters, e.g. <math>F(x)</math>, or <math>F_X(x)</math>.<ref name=":0" />
* [[Survival function]]s or complementary cumulative distribution functions are often denoted by placing an [[overbar]] over the symbol for the cumulative:<math>\overline{F}(x) =1-F(x)</math>, or denoted as <math>S(x)</math>,<ref name=":0" />
*In particular, the pdf of the [[standard normal distribution]] is denoted by φ(''z''), and its cdf by Φ(''z'').
*Some common operators:<ref name=":0" />
:* E[''X''] : [[expected value]] of ''X''
:* var[''X''] : [[variance]] of ''X''
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* X is independent of Y is often written <math>X \perp Y</math> or <math>X \perp\!\!\!\perp Y</math>, and X is independent of Y given W is often written
:<math>X \perp\!\!\!\perp Y \,|\, W </math> or
:<math>X \perp Y \,|\, W</math><ref name=":0" />
* <math>\textstyle P(A\mid B)</math>, the ''[[conditional probability]]'', is the probability of <math>\textstyle A</math> ''given'' <math>\textstyle B</math>,<ref
==Statistics==
*Greek letters (e.g. ''θ'', ''β'') are commonly used to denote unknown parameters (population parameters).
*A tilde (~) denotes "has the probability distribution of".
*Placing a hat, or caret, over a true parameter denotes an [[estimator]] of it
*The [[arithmetic mean]] of a series of values ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub> is often denoted by placing an "[[overbar]]" over the symbol
*Some commonly used symbols for [[Sample (statistics)|sample]] statistics are given below:
**the [[sample mean]] <math>\bar{x}</math>,
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**the population [[Pearson product-moment correlation coefficient|correlation]] ''ρ'',
**the population [[cumulant]]s ''κ<sub>r</sub>'',
*<math>x_{(k)}</math> is used for the <math>k^\text{th}</math> [[order statistic]],<ref name=":0" /> where <math>x_{(1)}</math> is the sample minimum and <math>x_{(n)}</math> is the sample maximum from a total sample size ''n''.
==Critical values==
The ''α''-level upper [[critical value]] of a [[probability distribution]] is the value exceeded with probability α, that is, the value ''x''<sub>''α''</sub> such that ''F''(''x''<sub>''α''</sub>) = 1 − ''α'' where ''F'' is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:<ref name=":0" />
*''z''<sub>''α''</sub> or ''z''(''α'') for the [[standard normal distribution]]
*''t''<sub>''α'',''ν''</sub> or ''t''(''α'',''ν'') for the [[Student's t-distribution|''t''-distribution]] with ν [[Degrees of freedom (statistics)|degrees of freedom]]
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*'''a.e.''' [[almost everywhere]]
*'''a.s.''' [[almost surely]]
* '''cdf''' [[cumulative distribution function]]<ref name=":0" />
* '''cmf''' [[cumulative mass function]]
*'''df''' [[degrees of freedom (statistics)|degrees of freedom]] (also <math>\nu</math>)<ref name=":0" />
*'''i.i.d.''' [[Independent and identically distributed random variables|independent and identically distributed]]<ref name=":0" />
*'''pdf''' [[probability density function]]<ref name=":0" />
*'''pmf''' [[probability mass function]]<ref name=":0" />
* '''r.v.''' [[random variable]]<ref name=":0" />
* '''w.p.''' with probability; '''wp1''' [[with probability 1]]
* '''i.o.''' infinitely often, i.e. <math> \{ A_n\text{ i.o.} \} = \bigcap_N\bigcup_{n\geq N} A_n </math>
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==References==
<references />
== Bibliography ==
*{{Citation| title=Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation| first1=Max|last1=Halperin |first2=H. O. |last2=Hartley |first3=P. G.|last3=Hoel | journal=The American Statistician| volume=19 |year=1965 | pages=12–14 | issue=3| doi=10.2307/2681417 | jstor=2681417}}
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