# Some maths and statistics questions

humy
Science 13 Jun '15 12:23
1. 13 Jun '15 12:2310 edits
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:

Normal distribution;
Exponential distribution;
binomial distribution;
hypergeometric distribution;
multinomial distribution;
Bernoulli distribution,
Poisson distribution;
uniform distribution

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.
2. Soothfast
0,1,1,2,3,5,8,13,21,
15 Jun '15 06:501 edit
Hi, Cav. 😉

My background in probability theory is not extensive, though I have studied it rigorously in the past. That is to say, I've studied it at the measure-theoretic level involving Lebesgue integration and Borel measurable functions. Well anyway...

So I understand a random variable X to be a Borel measurable function from a probability space (Ω,ℱ,P) to the set of real numbers ℝ. Any such X has an associated distribution function F : ℝ → [0,1] given by

F(x) = P({ω : X[ω] ≤ x})*

for all x ∈ ℝ. And the expected value of X? That's given as an integral over Ω:

E(X) = ∫ X dP.

This Lebesgue integral with respect to the probability measure P may indeed fail to exist, by which is meant it doesn't equal an "extended" real number (i.e. a real number or ±∞ ). I'm not aware of any nice "named" probability distributions for which this is the case. I mean, I imagine a random variable with some wonky distribution that results in an undefined expected value isn't of much practical value, though I may be wrong.

* I'm writing square brackets around ω, instead of parentheses, to avoid "smiley face syndrome".
3. 15 Jun '15 16:384 edits
Originally posted by Soothfast
Hi, Cav. 😉

My background in probability theory is not extensive, though I have studied it rigorously in the past. That is to say, I've studied it at the measure-theoretic level involving Lebesgue integration and Borel measurable functions. Well anyway...

So I understand a random variable X to be a Borel measurable function from a probability space ...[text shortened]... * I'm writing square brackets around ω, instead of parentheses, to avoid "smiley face syndrome".
Arr so, judging purely by that, in answer to my question 1;
there is a known type of probability distribution that has no mean average, but it most likely doesn't have a specific formal name; and I will say so in my book.

Thanks Soothfast 🙂
4. Soothfast
0,1,1,2,3,5,8,13,21,
15 Jun '15 19:56
Originally posted by humy
Arr so, judging purely by that, in answer to my question 1;
there is a known type of probability distribution that has no mean average, but it most likely doesn't have a specific formal name; and I will say so in my book.

Thanks Soothfast 🙂
When considering discrete random variables the rather forbidding-looking Lebesgue integrals become nice, simple sums. There appears to be a nice example of a (discrete) random variable that has no expected value here:

http://mathoverflow.net/questions/159222/mean-of-i-i-d-random-variables-with-no-expected-value

It's less pathological than I thought it would be, and could conceivably even arise in applications.
5. Soothfast
0,1,1,2,3,5,8,13,21,
15 Jun '15 20:05
Ah -- check out the Cauchy distribution. It's not one you listed, but it is found often in textbooks and is of importance. A random variable with a Cauchy distribution has no expected value.
6. Soothfast
0,1,1,2,3,5,8,13,21,
15 Jun '15 20:291 edit
Originally posted by humy
3, can I have absolute unequivocal conformation by an expert here that, in mathematics, infinity divided by infinity is ALWAYS and necessarily total nonsense?
As you may know, if a limit in calculus comes out to ∞/∞, it may yet be possible (with an application of L'Hopital's Rule or some such) to resolve the limit as a definitive real number. I assume you're just talking about a "straight-up" division ∞/∞, in which case I can only say I've not personally seen any instance where it is given a working definition. But I wouldn't be surprised to hear there's some sub-subfield of mathematics where it's defined to be 1 to make theorems nicer.

I do know, for instance, that at the intersection of measure theory with integration theory it is common practice to define 0·∞=0, though in calculus and elementary analysis courses 0·∞ would normally be called indeterminate. The motivation for such a definition is -- as it always is -- a "nicer" construct with wider applicability.
7. 16 Jun '15 06:391 edit
Originally posted by Soothfast
Ah -- check out the Cauchy distribution. It's not one you listed, but it is found often in textbooks and is of importance. A random variable with a Cauchy distribution has no expected value.
Thank for that! I have just looked at it and see that it has some similarity to my distribution because it doesn't have a mean for a similar reason but my distribution is still radically different from it. I must study this Cauchy distribution in due course.
8. 16 Jun '15 06:41
Originally posted by Soothfast
As you may know, if a limit in calculus comes out to ∞/∞, it may yet be possible (with an application of L'Hopital's Rule or some such) to resolve the limit as a definitive real number. I assume you're just talking about a "straight-up" division ∞/∞, in which case I can only say I've not personally seen any instance where it is given a working definition. ...[text shortened]... ion for such a definition is -- as it always is -- a "nicer" construct with wider applicability.
actually, I didn't know some of that! You have just taught me something new.