A variable is modelled as a random variable, without reference to the question whether it is truly random in reality.

For example, when the outcome of a flip coin is modelled as a random variable, no one asks the question whether the process of coin flipping is "really random".

This process wouldn't be "really random" if it's true that had we (or some hypothetical entity) known all possible physical parameters, it would be possible to predict the outcome with 100% certainity. It would be truly random if even given all possible knowledge about the physical reality, there would still be uncertainty with regards to the results of the coin flip.

  1. Why does the question (whether the random variable represents a process that is really random) not matter at the modelling stage?
  2. At what stage does it matter?
  • $\begingroup$ Regarding question 2, my own belief is that this matters when one has to interpret the model and understand the actual relationship between the model and reality. At least at that stage this question has to come up. However, I could not come with any real example from research. $\endgroup$ – Sam Sep 27 at 13:10

I think it makes sense to think of the world of mathematics and the real world as separated in principle. However, firstly in order to make real use of mathematics, the world of mathematics and the real world have to be connected by interpretation, including addressing/checking whether the connection is appropriate. Secondly, mathematics was originally, and is mostly still now, set up in the way it is in order to be useful, i.e., having a connection to the real world in mind.

A "random variable" is a mathematical construct, and as such part of the world of mathematics and not of the real world. When handling random variables in the world of mathematics, indeed it doesn't matter whether this represents a "real random process", because such things do not exist in the world of mathematics. It is however called a "random variable" because the people who defined it had in mind to model real random processes with it.

The problem with the second question is that it is not clear whether what you call "real random process" actually exists. There are determinists around who believe that nothing is "really" random. There are also people who hold that probabilities do not model randomness in the real world, but rather the state of knowledge/uncertainty of an individual, or a scientific community as a whole. These people still use random variables when treating probability mathematically.

Even a frequentist, for whom probability models model processes existing in reality may concede that many real processes for which we use random variables are not really random, although most believe that some are (such as radioactive decay, or random sampling and randomisation of experiments if the researcher uses proper random numbers). Ultimately "objective randomness" cannot be confirmed by observation, so I'd rather say, if a model of a real process is used by a researcher, the researcher has to do two things in order to convince others that the model is useful:

(a) argue that nothing of the knowledge of the process invalidates randomness (such as convenience sampling where a proper random sample could be drawn),

(b) show that the data behave (in all relevant aspects) about as the model implies they should behave.

This will not secure that the real process is "really random", however it suggests that we can use the model in order to learn something about what goes on in the real world, e.g., make predictions, or simply statements like "data on the effect of homeopathy look like if they were generated by a random model according to which it is the same as placebo."

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  • $\begingroup$ (1) Can you give a simple example where the data does not behave as the model implies it should (2) Is it possible to give an example where the data does behave as the model implies it should behave, and where the process does not invlidate randomness? $\endgroup$ – Sam Oct 11 at 7:37
  • $\begingroup$ Consider any good statistics textbook on regression modelling. These books will discuss diagnostics of the model assumptions and will show examples where you can see from the residuals, say, that some model assumptions are violated - and positive examples where model assumption diagnostics were done and all looks fine. Examples cannot be "simple" because I'd need to show specific data and introduce specific model assumptions for this. $\endgroup$ – Lewian Oct 11 at 10:49

something random is something you cannot predict. so if you can somehow solve an equation that figures out which side a coin will land on before it lands on that side, it is not truly random. you can do this for every random number generating process, so that variable is random if you cannot figure out what it will equal. it seems as if you can't, so for you it is truly random.

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  • $\begingroup$ for you it is truly random: so you define truly as being subjective as if truth were subjective? $\endgroup$ – Richard Hardy Sep 28 at 6:11
  • $\begingroup$ no, I just define 'truly random' as something that can be different for different people, like the truth about their age, or their gender, or their hair style. $\endgroup$ – forever Sep 28 at 15:59
  • $\begingroup$ also, something random is something that is unpredictable, and different people can predict different things, so whether something is random is different for different people. $\endgroup$ – forever Sep 28 at 16:05
  • $\begingroup$ Your two comments address two different things. The first one does not imply truth is subjective, but the second one kind of does. Random and unpredictable are not entirely the same. What is random is hard to predict, but what is hard to predict may or may not be random. $\endgroup$ – Richard Hardy Sep 28 at 16:22
  • $\begingroup$ if you think random does not mean unpredictable, what do you think random means? $\endgroup$ – forever Sep 28 at 16:30

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