What is a random variable...

[69dodge]

Said sequence (infinite or otherwise) would not be a product of a random variable. It would be a sequence of random variables.

I think he intends each element of his sequence to be a definite value. Considered individually, there's nothing random about it. Only the sequence as a whole is considered random, because the various values "have nothing to do with each other".

I agree with you that this is not the usual definition of "random variable". I'd call such a sequence "an uncomputable sequence". (Every finite-length sequence is computable, which perhaps answers mijopaalmc's question.)

If I were planning to roll a die once and then destroy it, so that there is no possibility of it generating an infinite sequence, I'd still be perfectly happy to consider the number that will result from the single roll to be a random variable.

(I don't mean that after I roll, say, a 4, I consider "4" to be a random variable. I mean that before I roll anything, I consider "the number that I will roll" to be a random variable.)

 

[cyborg]

The simple problem I am trying to get across is that it is undecidable in general as to whether or not some variable is random when one attempts to decide that by induction across its output.

Any "random" finite sequence has a "non-random" representation and you can't make a decision about an infinite sequence.

Randomness is defined by what it is not, not what it is.

 

[DevilsAdvocate]

Here is the comment that spurred this thread:



My contention in the thread in which the above comment occurs is that evolution is mathematically random because not every individual of a given phenotype produces reproductively viable offspring by virtue of their possessing that specific phenotype. As I understand it, the above observable fact jibe really well with the concept of a probability measure designating how often an event will occur with respect to all existing events in the sigma-algebra of a probability space.

Now it is also possible that I am misunderstanding evolutionary biology or the application of probability theory to evolutionary biology, but I am primarily interested in checking my factual knowledge of probability theory in and of itself.

I may be off the mark, but it seems you are picking semantic nits here. You say “evolution is mathematically random” and it “jibe really well with the concept of a probability measure”. The “nit” is the difference between being random and appearing random (or being able to be described as random). As sol invictus said above, this difference goes deep into questions about what “random” is, which could be everything or nothing or something approaching everything or nothing or some other quantum equation. Those questions seem to be independent of the meaning of random that you use. A “mathematically random” or “probability measure” is a theoretical explanation, evaluation, or prediction, which is what the theory of evolution is.

So if you want to get technical on the term, you have to define what you mean by “random”. It could mean anything from a general sense of uncertainty to a theoretical concept of probability to an unknown quantum equation. It seems you mean a general sense of the term and looks like someone is setting you up to say you mean a “theoretical concept of probability” to knock down evolution theory as something that can only occur in theory and not in the actual physical world. Which is throwing the evolution baby out with the bath water by adding nits on the meaning of the term “random”.

 

[technoextreme]

Simply speaking, a random variable is a variable that can take on a random value - no great surprise. The values are, of course, controlled by the statistical parameters of the collection they are drawn from - a Gaussian distribution with mean zero and standard deviation of 1 will have values clustered within a short distance of 0 usually, with the occasional value farther away from zero.

A random variable is a function making your definition really bad.

The simple problem I am trying to get across is that it is undecidable in general as to whether or not some variable is random when one attempts to decide that by induction across its output.

What????? A random variable is a mapping of a probability space so that we can make our lives easier.

(I don't mean that after I roll, say, a 4, I consider "4" to be a random variable. I mean that before I roll anything, I consider "the number that I will roll" to be a random variable.)

Technically yes and technically no. You actually should know what the sigma algebra is before you can figure out if what you are describing is the random variable.