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I've read that to call something a random variable, that thing must be the result of a statistical experiment. So it got me thinking in which situations might we have an actual bias?

For example, medical diagnosis is an interesting case.

The people who go to the doctor all have the thing in common -> a mass, let's say. It might be confirmed or disconfirmed by imaging. But is it truly random if it's not quite based on chance? (i.e. the people that don't have the mass didn't come to the doctor)

Karolis Koncevičius
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edward84
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    A random variable is a mathematical construct, not the result of a statistical experiment. The distinction is an important one, because it helps avoid confusion. In particular, you are free to use random variables to model things that (arguably) aren't random at all, such as closed Newtonian physical systems or actions taken by people. This understanding ought to encourage you to ask *how well* a random variable might model a phenomenon, rather than whether it's possible or not. – whuber Aug 05 '21 at 18:48
  • In case I misunderstood the source that stated this. Here it is: https://stattrek.com/probability-distributions/probability-distribution.aspx ('When the value of a variable is the outcome of a statistical experiment, that variable is a random variable') – edward84 Aug 05 '21 at 19:01
  • So in P(X), the X is not a particular/specific variable/feature in a dataset but a mathematical construct representing continuous and discrete variables? And the small x (P(X=x) is an actual value from a discrete or continuous variable in a dataset? – edward84 Aug 05 '21 at 19:15
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    That source is a challenge to understand correctly. The quotation neither defines nor correctly describes a random variable: it's merely a heuristic (and not a very good one at that) intended to convey some sense of what a random variable is used for. Unfortunately, the notation "$P(X)$" can mean lots of different things, so I can't help you there. Many writers do try to distinguish random variables and the values they might have by using upper case for the former and lower case for the latter. – whuber Aug 05 '21 at 19:17
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    I should add that elsewhere on this site I have made a few attempts to describe random variables in relatively non-mathematical terms. One is at https://stats.stackexchange.com/a/54894/919. – whuber Aug 05 '21 at 19:19
  • Thanks @whuber . I will have a look and report my findings :) – edward84 Aug 05 '21 at 19:23
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/128309/discussion-between-edward84-and-whuber). – edward84 Aug 06 '21 at 17:24
  • @whuber I read your answer and thought about it, and then read and watched countless other videos from experts, including a famous MIT prof. Much of the confusion seems to be based on the type of experience a person has (theoretical vs. practical). This MIT prof loosely defined random variables as 'random quantities that result from an experiment'. Then he gave a more precise math definition that treated random variables as a function. – edward84 Aug 06 '21 at 22:09
  • So, if you listen to the first 5 minutes. He mentions the experiment thing twice and connects it to a random variable. https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041-probabilistic-systems-analysis-and-applied-probability-fall-2010/video-lectures/lecture-6-discrete-random-variable-examples-joint-pmfs/ – edward84 Aug 06 '21 at 22:13

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