Notation in probability and statistics: Difference between revisions

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==Probability theory==

{{Unreferenced section|date=March 2021}}

* [[Random variable]]s are usually written in [[upper case]] Roman letters, such as <math display="inline">X</math> or <math display="inline">Y</math> and so on. Random variables usually, in this context, usually refer to something in words such as a continuous variable, such as "the height of a subject" orfor a discretecontinuous variable, such asor "the number of cars in the school car park" for a discrete variable. They are not numerical values. For instance if <math>P(X\le x) </math> is written then it means, the probability that a particular realisation of a random variable (say, height or number of cars) is less than or equal to a particular value <math display="inline">x</math>. It is important that <math display="inline">X</math> and <math display="inline">x</math> are not confused into meaning the same thing. <math display="inline">X</math> is an idea, <math display="inline">x</math> is a value. They are not related because they are the same letter.

* Particular realisations 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'' 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.