Notation in probability and statistics: Difference between revisions

Revision as of 12:12, 12 February 2024

Revision as of 02:22, 27 February 2024

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==Probability theory==

{{Unreferenced section|date=March 2021}}

* [[Random variable]]s are usually written in [[upper case]] romanRoman 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.<ref>{{Cite web |date=2021-08-09 |title=Calculating Probabilities from Cumulative Distribution Function |url=https://analystprep.com/cfa-level-1-exam/quantitative-methods/calculating-probabilities-from-cumulative-distribution-function/ |access-date=2024-02-26}}</ref>

* 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>.

*<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]]).

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==Statistics==

{{Unreferenced section|date=March 2021}}

*Greek letters (e.g. ''&theta;'', ''&beta;'') are commonly used to denote unknown parameters (population parameters).<ref>{{Cite web |date=1999-02-13 |title=Letters of the Greek Alphabet and Some of Their Statistical Uses |url=https://lesn.appstate.edu/olson/EDL7150/Components/Other%20useful%20links/Greek%20Alphabet%20and%20Statistics.htm |access-date=2024-02-26 |website=https://les.appstate.edu/}}</ref>

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

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

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**the population [[Pearson product-moment correlation coefficient|correlation]] ''<math display="inline">\rho</math>'',

**the population [[cumulant]]s ''<math display="inline">\kappa_r</math>'',

*<math>x_{(k)}</math> is used for the <math>k^\text{th}</math> [[order statistic]], where <math>x_{(1)}</math> is the sample minimum and <math>x_{(n)}</math> is the sample maximum from a total sample size <math display="inline">n</math>.<ref>{{Cite web |title=Order Statistics |url=https://www.colorado.edu/amath/sites/default/files/attached-files/order_stats.pdf |access-date=2024-02-26 |website=colorado.edu}}</ref>

==Critical values==