# A numerical probability distribution with no median!

humy
Science 21 Sep '17 09:41
1. 21 Sep '17 09:415 edits
In my research, I have derived the equations for HUNDREDS of totally new probability distributions (new to statistics) all of which I intend to publish in my book.
I have discovered some have unusual properties.

But now I have discovered one that by far has the weirdest property of them all!
I have discovered so many new distributions that I had to invent a systematic set of naming protocols (which I will explain in my book) else naming all of them would be totally arbitrary and chaotic. Using that systematic set of naming protocols, I call this particular probability distribution "One_up_i_u_outy_xteed".

But when I tried to derive the equation for its median using the usual old tried and tested and trusted methods for finding the median of any distribution, I found that for certain (but not all) input values, the resulting median equation I derived seems to be complete gibberish! It seemed to be saying that the median can be not only as much as "+infinity" but actually OVER "+infinity" !!!
And yet random variable x for this distribution must be finite i.e. x cannot be infinity.

Initially I naturally thought I must be doing something wrong with the maths so I have spend the last week with increasing frustration trying to see what I was doing wrong with my maths. But now I discovered it isn't the maths that is wrong but the assumption that it HAS a median! Although the distribution DOES normalize for all allowed input values, (I have checked this thoroughly) its cumulative distribution function has some extremely subtly weird properties for some of those input values that is such as to render all quantiles meaningless and "undefined" including for its median!

But now, here is my question;

Is my discovery truly unique or are there any known numerical probability distribution already discovered that sometimes have no definable median? Example?
2. sonhouse
Fast and Curious
21 Sep '17 13:051 edit
Originally posted by @humy
In my research, I have derived the equations for HUNDREDS of totally new probability distributions (new to statistics) all of which I intend to publish in my book.
I have discovered some have unusual properties.

But now I have discovered one that by far has the weirdest property of them all!
I have discovered so many new distributions that I had to inven ...[text shortened]... al probability distribution already discovered that sometimes have no definable median? Example?
I forwarded your post to my son in law Gandhi, don't know if he will even respond, he is really busy with students. We'll see. If he does, he will undoubtedly want concrete examples.
If you recall, he has Phd in Statistical physics so that should be right up his alley.
3. 21 Sep '17 16:41
There are distributions with an undefined mean, the Cauchy distribution is a well-known example.

A distribution without a median doesn't make sense to me. Any normalizable function will have a median, and if it can't be normalized it isn't a probability density function.
4. sonhouse
Fast and Curious
21 Sep '17 19:19
Originally posted by @kazetnagorra
There are distributions with an undefined mean, the Cauchy distribution is a well-known example.

A distribution without a median doesn't make sense to me. Any normalizable function will have a median, and if it can't be normalized it isn't a probability density function.
You mean like a true random distrubution of particals like in a Bose Condensate? One would be hard pressed to find a mean there.
5. 21 Sep '17 21:00
Originally posted by @sonhouse
You mean like a true random distrubution of particals like in a Bose Condensate? One would be hard pressed to find a mean there.
The distribution of particles in a Bose-Einstein condendsate isn't (necessarily) random. In many cases, particularly in the case of dilute, weakly interacting BECs, you can compute an approximate many-body wave function that quite closely agrees with the experiment using the Gross-Pitaevskii equation. Such a many-body wave function typically has a well-defined mean.
6. 22 Sep '17 02:10
Originally posted by @kazetnagorra
The distribution of particles in a Bose-Einstein condendsate isn't (necessarily) random. In many cases, particularly in the case of dilute, weakly interacting BECs, you can compute an approximate many-body wave function that quite closely agrees with the experiment using the Gross-Pitaevskii equation. Such a many-body wave function typically has a well-defined mean.
You should write gibberish for the next Star Trek movie, "Spock Finds Nemo."
7. 22 Sep '17 07:375 edits
Originally posted by @kazetnagorra
A distribution without a median doesn't make sense to me. Any normalizable function will have a median, and if it can't be normalized it isn't a probability density function.
Well you would think so! I thought so!

This particular distribution is for a discrete random variable g that can is a natural number and is thus this is for a probability mass function.
It always normalizes for all allowed input values.

However, and this is a bit that had me confused for nearly over a week, for certain input values (other than g itself because it has several input values) , it didn't 'appear' to me to normalize (even though it does! ) because for those input values, instead of the output of the cumulative distribution function for g input always tending to 1 as g tends to +infinity, depending on the exact input values (other than g itself) , it either tended to 1 OR 1/2 or 1/3 or 1/4 ...etc which I naturally thought was nonsense! But then I noticed that for those input values (other than g itself) , the is the possibility of a new kind of 'outcome' introduced, if you can call that an 'outcome', of there being 'no outcome' because the population becomes unsamplable . This 'outcome' of unsamplable population has, depending in the input values a probability mass of either 0 or 1/2 or 2/3 or 3/4 etc and it is for an outcome that has no meaningful numerical value (not even "g=0" ) and thus doesn't show up within the cumulative distribution function radar thus explains why the cumulative distribution function for g input 1 as g tends to +infinity, depending on the exact input values (other than g itself) , either tended to 1 or 1/2 or 1/3 or 1/4 ...etc; suddenly it all makes sense!
To see how it normalizes, you need to ADD the limit of the cumulative to the probability mass of 'outcome' of unsamplable i.e. the probability of NO possible outcome!

But now, what is the median for those cases where the cumulative distribution function cannot output 1/2 (or over) for ANY allowed g input value! ?
-I would say this makes nonsense of the median.
8. 22 Sep '17 16:14
Originally posted by @freakykbh
You should write gibberish for the next Star Trek movie, "Spock Finds Nemo."
https://en.wikipedia.org/wiki/Gross%E2%80%93Pitaevskii_equation
9. 22 Sep '17 18:056 edits
Originally posted by @kazetnagorra
https://en.wikipedia.org/wiki/Gross%E2%80%93Pitaevskii_equation
perhaps he would think THAT is just all made up "gibberish" along with anything else he cannot understand?
Meanwhile, he continues to use his computer and other technology that wouldn't work if it all was just made up "gibberish".
This is what you get from people with delusional arrogance of believing they know better than everyone else and there are no people smarter than them that know things they don't; and he is by far not the only one here with this sadly extremely common arrogant delusion.
In contrast, I have no such delusion and, only in part because of this, have learned much from people a lot smarter than me.
10. sonhouse
Fast and Curious
22 Sep '17 23:01
Originally posted by @kazetnagorra
The distribution of particles in a Bose-Einstein condendsate isn't (necessarily) random. In many cases, particularly in the case of dilute, weakly interacting BECs, you can compute an approximate many-body wave function that quite closely agrees with the experiment using the Gross-Pitaevskii equation. Such a many-body wave function typically has a well-defined mean.
Meaning it has some kind of overall shape, like a disk or some such?
11. 23 Sep '17 08:43
Originally posted by @sonhouse
Meaning it has some kind of overall shape, like a disk or some such?
Yes - mostly depending on whatever is containing the BEC.
12. 23 Sep '17 21:46
Originally posted by @sonhouse
I forwarded your post to my son in law Gandhi, don't know if he will even respond, he is really busy with students. We'll see. If he does, he will undoubtedly want concrete examples.
If you recall, he has Phd in Statistical physics so that should be right up his alley.
no need to forward. its all there in the akashic records. Ramanujan could easily tap in. you should too
13. 23 Sep '17 22:54
I'm not sure how you can have a finite group of numbers and not be able to find a median.
14. sonhouse
Fast and Curious
24 Sep '17 00:27