Thanks for talking yourself into agreeing with my point:
Even for an expected value to make sense, you don't need a 'number'; ie you don't need a field, and you don't even need a ring. You can use a less restricted structure.
> It is absurd to think about what is the expected value of a random experiment that produces the words "red", "green" and "blue" with various probabilities.
Why is it absurd? It's perfectly possible to define the result over a suitable 'free' structure. (In fact you can always do that, even for 'numbers' and then later collapse that free structure into something concrete.)
Btw, it's perfectly possible to define some weighted average of colours, if you wanted to. But that's about as relevant as the different not-quite-multiplications you brought up.
> It doesn't exist as a category, not due to a calculation problem. I.e. it's "not even undefined".
Free algebras are perfectly well studied structures in math. They 'exist' just as much as anything else in math does. And, by definition, they have all the right properties we need to define the expected value.
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> You can multiply vectors together in 2D (complex numbers) and 3D (cross product). Also 4D (quaternions, non-commutatively).
Those operations are often called 'multiplication', just like we often call any random group operation 'multiplication'. But there's no vector multiplication you can define in general (for all number of dimensions) that would give you a field or even just a ring. So they aren't really the kind of multiplication we need.)
Right; that needs to be better articulated. For the Cauchy distribution, we can meaningfully explore whether it has an expected value. We can write down the expression.
If it is not absurd to explore whether the distribution has an expected value (even though it may turn out that it isn't numerically defined), the distribution is of a random variable.
It is absurd to think about what is the expected value of a random experiment that produces the words "red", "green" and "blue" with various probabilities.
It doesn't exist as a category, not due to a calculation problem. I.e. it's "not even undefined".
> Eg you can't multiply vectors.
You can multiply vectors together in 2D (complex numbers) and 3D (cross product). Also 4D (quaternions, non-commutatively).
You can multiply vectors in any number of dimensions by scalars, which is all we need for averaging a bunch of vectors, with weights.
But yes, vector fields are not fields though, so vectors are not numbers in every sense.