Notation in probability and statistics: Difference between revisions

Revision as of 18:17, 3 May 2023

Revision as of 16:39, 8 June 2023

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{{Unreferenced section|date=March 2021}}

* [[Random variable]]s are usually written in [[upper case]] roman letters: <math display="inline">X</math>, <math display="inline">Y</math>, etc.

* Particular realizations of a random variable are written in corresponding [[lower case]] letters. For example, <math display="inline">x_1,x_2, \ldots,x_n</math> could be a [[random sample|sample]] corresponding to the random variable '' <math display="inline">X''</math>. 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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*Some common operators:

:* <math display="inline">\mathrm{E}[X]</math> : [[expected value]] of ''X''

:* <math display="inline">\mathrmoperatorname{var}[X]</math> : [[variance]] of ''X''

:* <math display="inline">\mathrmoperatorname{cov}[X,Y]</math> : [[covariance]] of ''X'' and ''Y''

* 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