Random variable: Difference between revisions

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{{Short description|Variable representing a random phenomenon}}

{{Probability fundamentals}}

Definition:

A random variables is a mathematical function which maps outcomes from a sample data set of different outcomes to different numerical values in a corresponding range of numerical values or numbers.

Denotation:

It can be denoted by any capital letter, and the outcomes from the sample data set can be denoted by a regular letter, which represents a variable value, and any outcome contained in the sample data set

Example: F(X) = 6, in which

F stands for the mathematical function which maps the random variable x to different numbers in a range of possible numbers

The capital X here stand for a sample data set which consists of different outcomes of an experiment, and we use the regular or lowercase letter x with different numerical subscripts, such as x1, x2, x3, x4, x5, x6, ... to represent unknown outcomes before the experiment begins. For instance, an experiment of 6 times of rolling a die. We have 6 possible outcomes, which are 1,2,3,4,5,6. Since we don't know, in advance, the outcome of the experiment of the rolling of the die each time we roll it, we use a letter, like the regular letter x to represent the unknown outcome before rolling the die, and since each time, the outcome of the rolling of the die can be 1 of the 6 possible numbers or numerical values mentioned above, we define the letter as a variable to mean that the value which it represents can change or vary each time when we roll the die. The reason why we consider the variable as being random is that the outcome which it represents each time when we roll the die is random or spontaneous, without any plan nor interference.The outcome comes without any arrangement by anyone. It can be one of the 6 possible numbers. We can treat the sample data set X = {x1, x2,x3, x4, x5, x6} as a ___domain of possible inputs, and the corresponding set of numbers F(X) ={ 1,2,3,4,5,6} as the range of corresponding outputs

That's what actually a random variable means

It is a very simple mathematical concept when human mathematicians can use unambigous and simple language as the way how I do. Otherwise,it can become very confusing or complicated when explained with ambiguous and obscure languages and a bunch of nonsensical mathematical formulas

A '''random variable''' (also called '''random quantity''', '''aleatory variable''', or '''stochastic variable''') is a mathematical formalization of a quantity or object which depends on [[randomness|random]] events.<ref name=":2">{{cite book|last1=Blitzstein|first1=Joe|title=Introduction to Probability|last2=Hwang|first2=Jessica|date=2014|publisher=CRC Press|isbn=9781466575592}}</ref> The term 'random variable' can be misleading as it is not actually random nor a variable,<ref>{{Cite book |last=Deisenroth |first=Marc Peter |url=https://www.worldcat.org/oclc/1104219401 |title=Mathematics for machine learning |date=2020 |others=A. Aldo Faisal, Cheng Soon Ong |isbn=978-1-108-47004-9 |___location=Cambridge, United Kingdom |oclc=1104219401 |publisher=Cambridge University Press}}</ref> but rather it is a [[function (mathematics)|function]] from possible [[Outcome (probability)|outcomes]] (e.g., the possible upper sides of a flipped coin such as heads <math>H</math> and tails <math>T</math>) in a [[sample space]] (e.g., the set <math>\{H,T\}</math>) to a [[measurable space]] (e.g., <math>\{-1,1\}</math> in which 1 corresponding to <math>H</math> and −1 corresponding to <math>T</math>), often to the real numbers.