any examples of 'modeless' continuous probability distributions?

Originally posted by twhitehead
I looked it up and it turns out I am correct. The 'definite integral' requires a closed set:
https://en.wikipedia.org/wiki/Integral

open sets result in 'improper integrals', but such things do exist.

This is why probability theory is best studied using full-fledged measure theory and Lebesgue integration. The Lebesgue integral can be evaluated over open sets no problem. (The "definite integral" introduced in calculus is otherwise known as the Riemann integral.)

Originally posted by Soothfast
This is why probability theory is best studied using full-fledged measure theory and Lebesgue integration. The Lebesgue integral can be evaluated over open sets no problem. (The "definite integral" introduced in calculus is otherwise known as the Riemann integral.)

Are you able to assist with any of the other issues that are causing confusion?

Originally posted by humy

But you can know something about that probability.

No, you cannot. Not in this case! At least not without some additional information. Not every theory has a probability under all circumstances! That is one of the many things I have already proved out of my intense research into epistemology over the last year (proof still to be published ...[text shortened]... ends" to zero but isn't zero doesn't answer the simple question "what value it is if not zero?".

Suppose one has an ideal fluid, such that the equation of continuity is rigorously true. Then one can examine the density and the mass in some volume. Transferring your argument to this fluid, you have that the mass is undefined at a point. One way out is to posit atoms - in other words go for some sort of finitist approach and insist that there is a smallest possible unit of volume. Another approach is to do as you have done and claim that the probability of a specific result is not a well defined quantity. What I tend to favour is to argue that the a priori probability one calculates is for a range of values and that in practice we have a precision limit (we can't measure people's heights to a precision better than a millimetre) so that the uncertainly in the result means that in any measurement we are sampling more than one value for the random variable. Quantum theory gets round this type of problem since there's fundamental uncertainty in the time of a radioactive decay, for example, and so any claim that t was the time at which the decay happened comes with a Δt so that the event is smeared out in time. Assuming Δt is of the order of Planck's constant and therefore small one can then say the probability is Δt*exp(-λt). So I think you are justified in that nature does not allow events, which if quantum theory is rigorously true are probabilistic in nature (pace arguments about interpretation), to occur at an overly specific time or place.

Edits: The site seems to not like 32 bit unicode characters, I had to use a less fancy delta 🙁

Originally posted by twhitehead

Saying it "tends" to zero but isn't zero doesn't answer the simple question "what value it is if not zero?".
What is the sum of the series:
1/2 + 1/4 + 1/8 ....
It isn't 1. It only tends to 1. And asking 'what is the value if it isn't one?' doesn't give you licence to say 'it is therefore undefined'.[/b]

No because that '1' doesn't refer to a probability of anything, let alone a logical possibility! THAT is the deference! That 1 comes from just pure mathematics, so of course it can be valid!
It is an epistemological contradiction for something to be BOTH logically possible in the external world AND to have zero probability in the external world! So even if the maths assigns a logical possibility of something in the external world a 0 probability, that just means it has an undefined probability because the maths assumes it has a probability and gives a nonsense 0 probability thus indicating that the implicit assumption the maths makes that it HAS a probability is wrong and that it must actually have no probability (or at least no probability until if or when we have more information on it ).

Originally posted by twhitehead
Are you able to assist with any of the other issues that are causing confusion?

Soothfast:

I believe the issue that if resolved will put to an end to what I see as by far the most horrible worse bit of the confusion here in this thread is what is the answer to this:

Is the exponential distribution (i.e. as in https://en.wikipedia.org/wiki/Exponential_distribution ) a "continuous probability distribution"?

If answer "no", why not and yet why is, say, a continuous uniform distribution is a "continuous probability distribution"?
I mean what difference in property between the two distributions makes the latter a "continuous probability distribution" but not the former?

Originally posted by humy
No because that '1' doesn't refer to a probability of anything, let alone a logical possibility! THAT is the deference! That 1 comes from just pure mathematics, so of course it can be valid!
It is an epistemological contradiction for something to be BOTH logically possible in the external world AND to have zero probability in the external world! So even if the ...[text shortened]... ve no probability (or at least no probability until if or when we have more information on it ).

Another alternative, which I should have mentioned above, is to say that the probability of the random variable having a particular value is infinitesimal. It seems to fit the bill as they're defined to be smaller than any positive definite real number but non-zero so you don't have the logical nonsense of a possible result being impossible, while avoiding the necessity for atomism or relying on assumptions about shortest distance or time scales.

Originally posted by DeepThought
Another alternative, which I should have mentioned above, is to say that the probability of the random variable having a particular value is infinitesimal. It seems to fit the bill as they're defined to be smaller than any positive definite real number but non-zero so you don't have the logical nonsense of a possible result being impossible, while avoiding the necessity for atomism or relying on assumptions about shortest distance or time scales.

you wont believe this but I had already independently thought up all your alternatives months ago on this except this one.
I will give this one some thought. But my first thought on this is the exact numerical value of this "infinitesimal" probability would have to be undefined (I think you already implied this ) because as soon as you assign any particular finite non-zero value to it, no matter how incredibly small, then you would have an infinite number of these infinitesimal over a range of x that will sum to infinite probability, which is nonsense.
But, I ask myself, if each "infinitesimal" value is undefined, isn't that saying, in effect, the some thing as the probability of any one particular x value is undefined just like I did before? Or is there a subtle but fundamental difference here because I was not using the concept of "infinitesimals"? These are my initial thoughts.

I just had another thought! not only can we not assign each infinitesimal a specific none-zero finite value (you already implied this ), we must insist that they all have different unknown values with an infinite number of differing values! Else you will have an infinite number of infinitesimals over a finite range of x with the same non-zero probability and the sum of an infinite number of ANY non-zero equal probabilities is a nonsense infinite probably!

Originally posted by humy
can anyone give me an example of a continuous probability distribution that has no definable mode or modes?
I am note talking here about, say, a continuous uniform distribution, which arguably has an infinite number of modes along a 'plateau' on its probability density function thus merely has no unique mode/modes. I am talking here about a continuous d ...[text shortened]... or another example of a 'modeless' continuous distribution but got absolutely nowhere.
Anyone?

Maybe the probability of getting sucked into a black hole as you get closer and closer to it?

The craters on the moon and other bodies would possibly fit if they were all caused by impact. One of the accepted theories is that the reason most of the craters are round is because they explode on impact creating a symetrical crater instead of elongated craters one would expect from impacts at different angles. The problem with this is that the craters are not round but hexagonal. An explosion would not create a hexagonal crater. The electric universe theory says that the craters are mostly caused by electric discharge. It has been proven electric discharge does create hexagonal craters. If this theory is false then the probability distribution of impact angle is as modeless as the human behavior example posted earlier.

Originally posted by humy
It is an epistemological contradiction for something to be BOTH logically possible in the external world AND to have zero probability in the external world!

Well that obviously depends on your definition of 'probability'.

So even if the maths assigns a logical possibility of something in the external world a 0 probability,
The maths doesn't actually assign a probability of 0. It assigns an infinitesimally small probability. But it isn't undefined.

The series I quoted above can we written as a sequence of partial sums. The sequence of partial sums does not have 1 as a member. Similarly the probability can be thought of as being open at zero ie it is everything upto but not including zero. But it is not undefined. The probability is infinitesimally small and that is useful and actionable information and is neither useless nor undefined.

As DeepThought points out, in real life mathematically perfect probabilities of this nature simply don't exist in real life, and your function would not be allowed anyway.

I think the main confusion is that you are dealing with infinities but refusing to accept them. That thinking is what gives people problems with Zenos paradox. Your declaration that it is 'undefined' is similar to claiming that a runner can never finish a race.

Originally posted by twhitehead
Well that obviously depends on your definition of 'probability'.

[b]So even if the maths assigns a logical possibility of something in the external world a 0 probability,
The maths doesn't actually assign a probability of 0. It assigns an infinitesimally small probability. But it isn't undefined.

The series I quoted above can we written as a s ...[text shortened]... declaration that it is 'undefined' is similar to claiming that a runner can never finish a race.[/b]

Similarly the probability can be thought of as being open at zero ie it is everything upto but not including zero.

if that probability is an 'infinitesimal' i.e. a probability (NOT probability density, which isn't a true probability ) of just one value of continuous random variable x,"everything" means, say BOTH value a and value b where a≠b (and it doesn't matter how 'close' a and b are to zero ). But that will be a contradictory probability because, a probability, if it exists for something, cannot have more than one value. Thus if a said 'probability' both equals a and b where a≠b then that indicates it is undefined.

I think the main confusion is that you are dealing with infinities but refusing to accept them. That thinking is what gives people problems with Zenos paradox. Your declaration that it is 'undefined' is similar to claiming that a runner can never finish a race.

No it isn't! Because 1/2 + 1/4 + 1/8 .... is in this case not referring to a probability nor probabilities.

Probability has to obey slightly different rules to avoid certain epistemological contradictions which don't apply to just distances, speeds and time intervals alone without any relation to probability. For example, you can without a contradiction refer to a logical possibility, NOT to be confused with a probability, of a time interval between two events being exactly 0 ; no problem! But, as soon as you refer to a probability of a time interval between two events being exactly 0 when it is also logically possible for the time interval between two events being exactly 0, NOW you have a problem!

Originally posted by humy
,"everything" means, say BOTH value a and value b where a≠b

I used careless language. What I mean is the probability is the smallest value in the open set open at zero. So its like the maximum value in your exponential graph isn't at zero but it is infinitesimally close to zero. We can all agree I hope that the maximum value is unique and that it exists and that it is not undefined. But we cannot rightly say it is at zero, nor can we say it is not. Therefore we must couch our language in limits. What we shouldn't do is say 'it is undefined and there is no meaningful information about it'. Clearly there is meaningful information and clearly the probability of any given point occurring is infinitesimally small.

No it isn't! Because 1/2 + 1/4 + 1/8 .... is in this case not referring to a probability nor probabilities. Probability has to obey slightly different rules to avoid certain epistemological contradictions which don't apply to just distances, speeds and time intervals alone without any relation to probability.
I think you will find that Zenos paradox also has certain epistemological contradictions that have confused people for centuries. Ultimately both situations come down to dealing with infinities and a rejection of infinities leads to apparent contradictions and confusion. You need to come to terms with infinities rather than throwing your hands up in the air and saying that the runner cannot cross the finish line.

Originally posted by twhitehead
Well that obviously depends on your definition of 'probability'.
...

exactly!

Clearly there is meaningful information and clearly the probability of any given point occurring is infinitesimally small.

I think that should be "...any given point occurring, within some arbitrary defined infinitesimally small range of x values, is infinitesimally small"
Only then do I think it has 'meaningful information' ( about its probability ) .

Originally posted by humy
exactly!

You are insisting on holding on to a definition that only deals with finite numbers but trying to use it on infinite sets.

I could be wrong but my current thinking is:

If we are dealing just with pure mathematics, I assume making a special distinction between an unreachable limit and a reachable limit would generally be petty superfluous. But, if we are dealing not with just pure mathematics but specifically dealing with probabilities relevant in the physical world, then the distinction between unreachable limit and reachable limit may be necessary to avoid certain non-mathematical epistemological contradictions even though absolutely no mathematical contradiction or any other kind of mathematical 'problem' may result from failure to make such a distinction.

Let me put it this way:

a probability; P(x) = 0 can simultaneously be, without any contradiction, BOTH mathematically correct, i.e. make perfect mathematical sense, AND epistemological wrong, i.e. total nonsense!
So if I say "P(x) = 0 is wrong!", it is no good you asking; "Well mathematically PROVE it wrong then! Show us the maths that shows that it is wrong! WHY is the mathematical function for that invalid?"
-the problem with that is that the source of the error isn't the maths! The maths is all just fine!
You must look beyond the maths and ask "yes, the maths says this and I have no argument with the maths. But what does that really mean for what is in the REAL physical world? Does it really make sense for the REAL world?"