Notation in probability and statistics: Difference between revisions

Revision as of 10:57, 4 January 2023

Revision as of 00:19, 13 February 2023

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* [[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, ''x''₂, …, ''x''_''n'' 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”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>.

*<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”"or" in this case means [[inclusive or|one or the other or both]]).

*[[sigma-algebra|&sigma;-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>.