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I am learning about random variables with all of their different types of distributions for discrete and continuous types. However, before knowing about random variables, I am not sure what would be a variable in statistics which is not random? Is not every variable by chance and everything already a random variable? What would be the "opposite" of a random variable i.e. some variable which is NOT to happen by chance?

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  • $\begingroup$ what is the context of this question? Maybe you can write a sentence and leave ___ blank where you wanna use the concept you are looking for? $\endgroup$ – aaaaa says reinstate Monica Nov 7 '19 at 21:11
  • $\begingroup$ Just to be clear, the term "random variable" in this context is to be considered as a single term, not a composed term "random" + "variable". Thus, it makes no sense to ask what a "non-random" + "variable" is... $\endgroup$ – Daniel Robert-Nicoud Nov 7 '19 at 22:32
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A random variable which is not actually random, and doesn't change by chance, is by definition a constant. But, it is still a RV. Since the RV definition is a superset of constant RV definition, I believe there is no conceptual opposite.

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    $\begingroup$ It doesn't have to be constant, and can be modeled as random. But even if it is constant, it's still a RV by definition. How you model your variables usually depends on you. $\endgroup$ – gunes Nov 7 '19 at 7:26
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    $\begingroup$ For the coin flip, you'd say the outcome is a V~Bernoulli(P) where V (in {Heads, Tails}) is a Bernoulli random variable with a constant parameter P=0.5. (As @gunes mentions, a constant is technically an RV will all its probability mass concentrated at a single point). Sometimes, like for example if you don't know if the coin is biased or not, you might even treat P (in [0, 1]) as an RV. Typical textbook problem is to compute the distribution over P given the evidence of seeing the outcomes of N coin-flips. $\endgroup$ – Peter Nov 7 '19 at 16:55
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    $\begingroup$ -1 A variable which is both deterministic and non-constant is not a random variable. For example, $x_{1}=1,x_{2}=2,\dots,x_{n}=n$ is a trivial non-random variable. A constant is, by definition, non-variable. $\endgroup$ – Alexis Nov 8 '19 at 0:16
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    $\begingroup$ @Alexis, I appreciate the comment. I also think that it's misleading to refer it as just variable. Reworded. $\endgroup$ – gunes Nov 8 '19 at 6:06
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    $\begingroup$ Your revision still neglects that variables can be non-random without being constant. Another example: $x_1 = 1, x_{2} = 0, x_{3} = 1, \dots$ (i.e. $x$ is an indicator variable for the oddness of the index). $\endgroup$ – Alexis Nov 8 '19 at 15:59
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One thing that might be worth noting is that in the formal definition, a random variable is a function -- in particular, a measurable function $X: \Omega \to E$ from a set of possible outcomes $\Omega$ (which is in fact a probability space -- more here) to a measurable space $E$.

Along the same lines of @gunes answer (+1), it doesn't quite make sense to discuss the opposite of a function -- you could say it's a constant, but how would you consider a function such as $f(x) = 0$? Is it "more" constant than other functions? It's a bit like comparing apples and oranges, since functions and scalars are very different types of objects.

I think your question is more around the use of the word "variable", which can be a bit confusing. For instance, in algebra you might encounter a problem such as "Find the roots of the equation $x^2-9=0$". Here, $x$ is a "variable", but it takes on a deterministic value (namely, $ x = \pm 3$ ) and can really be considered scalar since $x \in {\Bbb R}$. There's no presumption of it representing a relationship between some event and an associated probability, so it's not considered a random variable.

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A non-random variable is generally called a Constant. But constants are not really the opposite of random variables, in the same way integers are not the opposite of real numbers - they're a subset.

A constant is just a random variable with all it's probability mass concentrated at one point. (i.e. it has a Dirac-delta function for probability distribution)

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