Notation in probability and statistics: Difference between revisions

Revision as of 13:45, 27 April 2023

Revision as of 13:53, 27 April 2023

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{{Unreferenced section|date=March 2021}}

* [[Random variable]]s are usually written in [[upper case]] roman letters: ''X'', ''Y'', etc.

* Particular realizations of a random variable are written in corresponding [[lower case]] letters. For example, ''x''₁,math ''x''<sub>2</subdisplay="inline">x_1,x_2, …\ldots, ''x''<sub>''n''x_n</submath> 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.

*<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>.

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*A tilde (~) denotes "has the probability distribution of".

*Placing a hat, or caret, over a true parameter denotes an [[estimator]] of it, e.g., <math>\widehat{\theta}</math> is an estimator for <math>\theta</math>.

*The [[arithmetic mean]] of a series of values ''x''₁,math ''x''<sub>2</subdisplay="inline">x_1,x_2, ...\ldots, ''x''<sub>''n''x_n</submath> is often denoted by placing an "[[overbar]]" over the symbol, e.g. <math>\bar{x}</math>, pronounced "''<math display="inline">x''</math> bar".

*Some commonly used symbols for [[Sample (statistics)|sample]] statistics are given below:

**the [[sample mean]] <math>\bar{x}</math>,

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==Critical values==

{{Unreferenced section|date=March 2021}}

The ''&alpha;''-level upper [[critical value]] of a [[probability distribution]] is the value exceeded with probability &alpha;, that is, the value ''x''_''&alpha;'' such that ''F''(''x''_''&alpha;'') =&nbsp;1&nbsp;&minus;&nbsp;''&alpha;'' where ''F'' is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:

*''z''<submath display="inline">''&z_\alpha;''</submath> or ''<math display="inline">z''(''&\alpha;'')</math> for the [[standard normal distribution]]

*''t''<submath display="inline">''&t_{\alpha;'',''&\nu;''}</submath> or ''<math display="inline">t''(''&\alpha;'',''&\nu;'')</math> for the [[Student's t-distribution|''t''-distribution]] with &<math display="inline">\nu;</math> [[Degrees of freedom (statistics)|degrees of freedom]]

*<math>{\chi_{\alpha,\nu}}^2</math> or <math>{\chi}^{2}(\alpha,\nu)</math> for the [[chi-squared distribution]] with &nu; degrees of freedom

*<math>F_{\alpha,\nu_1,\nu_2}</math> or <math display="inline">F(&\alpha;,''&nu;''₁\nu_1,''&nu;''<sub>2\nu_2)</submath>) for the [[F-distribution]] with ''&nu;''<submath display="inline">1\nu_1</submath> and ''&nu;''<submath display="inline">2\nu_2</submath> degrees of freedom

==Linear algebra==