I have some burning questions but first; to put this into context:
I am currently doing some epistemological research and am writing a book, which I estimate will be published in about one years time, about a new system of logic called “entropic logic” that I am developing. I believe I have a responsibility to be careful in this post (and all others ) not to give away any of the most critical details of my research until publication else I risk falling victim to plagiarism.
Using this “entropic logic”, I have already deduced a new kind of probability as well as a whole now class of probability distributions, which I call “cav distributions” (for reasons I cannot reveal here ) that will be totally new to the science of statistics.
But, unless I am mistaken, and this is one of the things I really want to check here, I have discovered that there is a small subclass of these “cav distributions” that, to my surprise, have the eccentric property that they are such that they have the surprising property that they have no expected value i.e. mean average! And I am not talking here about the mean being merely unknown nor am I talking about a likelihood function. I am talking about there not existing ANY mean of the distribution! Whether a known one or an unknown one!
Now, with all the other [conventional] distributions I have so far studied, they all have a expected value i.e. mean average. These include:
So, here is my first 2 questions:
1, Is there any known type of probability distribution already known to have no mean average and, if so, what is it called? (if not, then I really may have discovered a new kind of distribution extremely unlike any conventional one! )
2, is there any known reason to think that a probability distribution must necessarily have a mean average even if it is an unknown one? (if so, then I must somehow have gone drastically wrong here with my reasoning and/or maths ) and, If so, what is that reason?
Now, the reason why I conclude that the distribution I discovered has no mean is because, after trying a numerical approach to find the mean and getting complete gibberish outputted out of my computer program, as for as I can now deduce, the only way to calculate the mean (if it exists ) is by dividing infinity by infinity so that, in this case;
Mean average = infinity / infinity.
Which I assume to be complete mathematical nonsense! I think I am pretty much ~99% sure of this But, just to make sure;
3, can I have absolute unequivocal conformation by an expert here that, in mathematics, infinity divided by infinity is ALWAYS and necessarily total nonsense?
But, it has occurred to me that, perhaps, just because one cannot calculate the mean average in this case, doesn't mean it doesn't exist!? Perhaps it could be like trying to find the fractions i.e. which whole number divided by which whole number, equals Pi! Just because you cannot calculate Pi exactly that way, doesn't mean there is no Pi! But perhaps that isn't quite the correct analogy here because, in this case, there appears to be no way of even giving an estimate of it!
So, here is my last question:
4, Could there be a mean average of something that is impossible to calculate, not even approximately, but nevertheless exist!? (Or is that just total nonsense? )
P.S. just as a matter of interest, I have also discovered a "cav distribution" which is such that, no matter how large the sample space, you must always necessarily assume, as the best-estimate of the mean average, the mean average of the actual distribution to be exactly the mean average of that in the sample space plus 1/2. I am also having difficulty getting my head around that "plus 1/2" part. But I have thoroughly checked that in several different ways and, despite it intuitively still seeming wrong to me, that is definitely correct and for extraordinary subtle reasons I would hate to have to explain.