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→Probability theory: the probability that a particular realisation of a random variable (e.g., height, number of cars, or bicycle colour), ''X'', would be equal to a particular value or category (e.g., 1.735 m, 52, or purple), <math display="inline">x</math>. |
→Probability theory: Clearly they are related, but they do not have identical meanings. |
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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, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable. They do not represent a single number or a single category. For instance, if <math>P(X = x) </math> is written, then it represents the probability that a particular realisation of a random variable (e.g., height, number of cars, or bicycle colour), ''X'', would be equal to a particular value or category (e.g., 1.735 m, 52, or purple), <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.
* 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, <math>X</math> is a random variable that is a function of <math>\omega</math> (i.e., it depends upon <math>\omega</math>), and <math>\omega</math> is some outcome of interest within the ___domain specified by <math>\Omega</math> (say, a particular height, or a particular colour of a car). <math>\Pr(A)</math> notation is used alternatively.
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