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Montgolfière (talk | contribs) Mixture distributions generally do have densities |
Giving an accurate, unambigous and simple definition with a well known example Tags: Reverted Mobile edit Mobile web edit |
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{{Short description|Variable representing a random phenomenon}}
{{Probability fundamentals}}
Definition:
A random variables is a mathematical function which maps outcomes of an from a sample data set 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
F stands for the mathematical function which maps the random variable x to different numbers in a range of possible numbers
x here is an out come from a sample data set of different outcomes, such as that of an experiment 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, we use a letter, like the 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.
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.
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