31 Jan 16
11 edits
twhitehead
I at last have asked a real maths expert at a proper maths forum about one of the things we were talking about and I got a reply that I believe will clear up what I personally see as, for me, the worst part of the confusion here:
I asked I Ask Clyde Oliver a Question about Probability & Statistics
at: http://www.allexperts.com/user.cgi
This question:
Is the exponential distribution (i.e. as in https://en.wikipedia.org/wiki/Exponential_distribution ) a "continuous probability distribution"?
...
....
... I had always thought the word "continuous" in the term "continuous probability distribution" was indicating that the "probability distribution" is FOR a "continuous random variable". ......
...And, contrary to what I always thought, he insists that the exponential distribution is NOT a "probability distribution" because it is "not continuous". ....
...
(all superfluous parts of my question deleted )
And then maths expert on Probability & Statistics answered:
First: It says right in the introduction to the Wikipedia article that this is a continuous distribution. So your answer is right there. You are correct. Further, you are correct entirely. Your definition is entirely correct:
I had always thought the word "continuous" in the term "continuous probability distribution" was indicating that the "probability distribution" is FOR a "continuous random variable"
That is a true, correct, and essentially complete characterization of what it means to be a continuous probability distribution.
Now, to elaborate.
A continuous probability distribution is one in which the associated random variable is continuous (i.e. not discrete).
See:
If a random variable is a continuous variable , its probability distribution is called a continuous probability distribution. [1]
A continuous distribution describes the probabilities of the possible values of a continuous random variable. A continuous random variable is a random variable with a set of possible values (known as the range) that is infinite and uncountable. [2]
A random variable X taking values in set S is said to have a continuous distribution if P(X=x)=0 for all x∈S.[3]
When you work with the normal distribution, you need to keep in mind that it's a continuous distribution, not a discrete one. A continuous distribution's probability function takes the form of a continuous curve, and its random variable takes on an uncountably infinite number of possible values. This means the set of possible values is written as an interval, such as negative infinity to positive infinity, zero to infinity, or an interval like [0, 10]...[4]
A continuous probability distribution is a probability distribution that has a cumulative distribution function that is continuous.[5]
All of the above definitions are satisfied by the exponential distribution except the one from "Probability for Dummies" (second to last). I have underline the incorrect part of their definition. Perhaps "Probability for Dummies" was written by dummies? It is possible they meant to state what the last definition (from Wikipedia) says, which is that the cumulative distribution function is continuous. (Or perhaps simply that the domain of the probability distribution function is continuous, i.e. the random variable is continuous.)
The function that describes the probability distribution function of the exponential distribution is not a continuous function. It is zero for values x<0 and λe^(-λx) for x≥0, causing a jump discontinuity in the function.
However, that is not relevant to the definition of "continuous distribution". The random variable in question takes on values on the real number line, which means it is a continuous random variable. It does not matter if the function that defines the PDF is continuous, especially since the CDF (which is really more important in that sense) will still be continuous.
So I was right all along about that:
An exponential distribution IS a continuous probability distribution.
And the fact that it is not a continuous function has nothing to do with it.
And, by implication, my OP distribution is also a continuous probability distribution.
( + I personally learned something new by what he said there about continuous functions )
I might ask him another question if confusion persists here.
01 Feb 16
1 edit
Originally posted by humySo was I. ie I was right that your example did not match the Wikipedia definition. I am happy to accept that the Wikipedia definition is wrong.
So I was right all along about that:
I still maintain however that the zero in your function is not part of the sample space and the probability at zero is undefined.
I believe the mode should be considered to be at zero despite it not being in the sample space. I also believe that no special name is required.
01 Feb 16
6 edits
Originally posted by twhiteheadExactly where in his post does he say/imply my example "did not match the Wikipedia definition"? He clearly said/implied no such thing and indirectly implied that it would match.
So was I. ie I was right that your example did not match the Wikipedia definition. I am happy to accept that the Wikipedia definition is wrong....
here is the relevant part of his quote again:
... The function that describes the probability distribution function of the exponential distribution is not a continuous function. It is zero for values x<0 and λe^(-λx) for x≥0, causing a jump discontinuity in the function.
However, that is not relevant to the definition of "continuous distribution". The random variable in question takes on values on the real number line, which means it is a continuous random variable. It does not matter if the function that defines the PDF is continuous, especially since the CDF (which is really more important in that sense) will still be continuous.
Thus he is clearly saying An exponential distribution IS a continuous probability distribution.
And the fact that it is not a continuous function has nothing to do with it.
( And, by implication, my OP distribution is also a continuous probability distribution.)
How is this not true that he is saying this?
I now have had a real maths expert, a professor specifically with real credentials in statistics, that says an exponential distribution is a continuous probability distribution ( + by implicit implication so is my distribution ), exactly like I thought from wiki but all other sources including one of my university courses. Would you say he, a real expert on the subject, is also wrong?
As for your other assertions; I will ask him some more questions in due course to find out if you are also wrong about them.
01 Feb 16
1 edit
Originally posted by humyThe Wikipedia definition which I assume is what is labelled 1 though 5 in your post states that the probability function is continuous in 4 and again repeated in 5.
Exactly where in his post does he say/imply my example "did not match the Wikipedia definition"? He clearly said/implied no such thing and indirectly implied that it would match.
However, that is not relevant to the definition of "continuous distribution"
Essentially saying that 4 and 5 are wrong.
The random variable in question takes on values on the real number line, which means it is a continuous random variable.
Agreed, with the understanding that it takes on values of a subset of the real number-line, not the whole number line. I would argue that defining a probability function to have a value of zero for sections of the number-line effectively excludes those sections from the sample space and the probability should be considered undefined at those points and not zero.
Thus he is clearly saying An exponential distribution IS a continuous probability distribution.
And the fact that it is not a continuous function has nothing to do with it.
Agreed. But that is not what Wikipedia says hence Wikipedia is wrong.
Would you say he, a real expert on the subject, is also wrong?
No, I have not disputed what he has said.
As for your other assertions; I will ask him some more questions in due course to find out if you are also wrong about them.
I am more than willing to be corrected.
01 Feb 16
1 edit
twhitehead
Before I decide what to ask him next, I would like you to answer this question so I can ask him about the exact thing that is causing the confusion here because it is still unclear to me exactly what that thing is:
Are there ANY circumstances which you would say a probability of something can be correctly said to be exactly 'zero' as opposed to 'undefined'?
If so, give me just one example of a probability distribution that gives a valid zero probability for something as opposed to a probability that is 'undefined' i.e. doesn't exist.
Or are you saying that, in conventional statistical terminology, ALL definable probabilities MUST be greater than zero? -i.e. NO exceptions!?
01 Feb 16
1 edit
Originally posted by humyNo. That would violate the definition of 'sample space'.
Are there ANY circumstances which you would say a probability of something can be correctly said to be exactly 'zero'?
Since your expert seems to have a very low opinion of Wikipedia (which I think is unjustified and elitists and has no place in mathematics), let me give a reference to a document from the University of Illinois. I have not checked the credentials of said university.
http://www.math.uiuc.edu/~kkirkpat/SampleSpace.pdf
The set of all the possible outcomes is called the sample space
of the experiment and is usually denoted by S.
A probability of exactly 'zero' would imply it is not a possible outcome and therefore not in the sample space.
01 Feb 16
2 edits
Originally posted by twhiteheadLet me rephrase the question:
No. That would violate the definition of 'sample space'.
Are there ANY circumstances which you would say a probability of something, REGARDLESS of whether or not you say that probability is part of a 'sample space', can be correctly said to be exactly 'zero'?
Let me put to you a simpler similar (or same? ) question:
Is there such thing as zero probability?
01 Feb 16
Originally posted by humyAll definitions about probability, including the definition of what it means to ask what the probability of an event is, start from the concepts of sample spaces and events. You are essentially asking about the probability of an event outside the sample space, but that is incoherent as the definition of 'event' is that it is within the sample space:
Let me rephrase the question:
Are there ANY circumstances which you would say a probability, REGARDLESS of whether or not you say that probability is part of a 'sample space', of something can be correctly said to be exactly 'zero'?
From the University of Illinois document again:
Any subset E of the sample space S is called an event.
Don't like my University of Illinois reference? Feel free to post a link to any other authoritative text book on the subject.
01 Feb 16
12 edits
Originally posted by twhiteheadwhere does it say there that you cannot have a "zero probability" of a considered hypothetical event that isn't inside the sample space i.e. that event hasn't/won't ever occur in reality i.e that conceived/imagined event doesn't exist?
All definitions about probability, including the definition of what it means to ask what the probability of an event is, start from the concepts of sample spaces and events. You are essentially asking about the probability of an event outside the sample space, but that is incoherent as the definition of 'event' is that it is within the sample space:
Fr ...[text shortened]... llinois reference? Feel free to post a link to any other authoritative text book on the subject.
According to you, IS there such thing as 'zero probability' at all!?
Is "zero probability" a self-contradiction!? Is THAT what you are really saying?
( because I don't know! )
01 Feb 16
1 edit
Originally posted by humyAn event is defined to be in the sample space. To ask about an event outside the sample space is incoherent.
where does it say there that you cannot have a zero probability of a considered hypothetical event that isn't inside the sample space
If you flip a two headed coin, what is the probability that a cow will jump over the moon? I am being incoherent.
According to you, IS there such thing as 'zero probability' at all!?
There is no such thing as truly zero probability. In continuous probability theory all discrete events have a zero probability but that is a 'limit zero' and not a 'true zero' ie they can happen, but won't 🙂
01 Feb 16
10 edits
Originally posted by twhitehead
An event is defined to be in the sample space. To ask about an event outside the sample space is incoherent.
If you flip a two headed coin, what is the probability that a cow will jump over the moon? I am being incoherent.
[b]According to you, IS there such thing as 'zero probability' at all!?
There is no such thing as truly zero probability. In ...[text shortened]... ro probability but that is a 'limit zero' and not a 'true zero' ie they can happen, but won't 🙂[/b]
If you flip a two headed coin, what is the probability that a cow will jump over the moon? I am being incoherent.
You would be 'incoherent' as in 'confusing' and 'idiotic' but I fail to see any 'deductive-logic-related incoherence' in that question as flipping " a two headed coin" merely makes no difference to "the probability that a cow will jump over the moon" i.e. it is irrelevant information thus making the question 'idiotic', but not contradictory.
It would be like me asking:
"if I am climbing up the walls stark raving mad, WHY do lions hunt in packs!? 😠"
The question would be 'idiotic', but still with no self-contradiction and perfectly answerable.
According to you, IS there such thing as 'zero probability' at all!?
There is no such thing as truly zero probability.
Oh you are making me go around and around in circles here to get an answer;
OK then, what is the difference between a "truly zero probability" (which you say doesn't exist) and a "zero probability" that is not "truly" "zero probability" ?
-you give me examples of the meaning of the latter but not the former.
... but that is a 'limit zero' and not a 'true zero' ie they can happen,
But that is exactly what I have been saying all along!
01 Feb 16
1 edit
Originally posted by humyIt is only answerable because 'why' is not defined in a way that depends on something to do with you climbing walls. Probability on the other hand is defined in reference to a sample space and holds no meaning outside that sample space. At best you can use the word in reference to a different sample space - which is what you actually did with the cow jumping over the moon. But it must be clear when you ask or answer it that you are talking in reference to a new sample space, whereas in your OP example you are not clear about that and in your 'zero probability' question you are not only not clear about it, but I believe getting confused about it. ie you are trying to maintain the reference to the sample space whilst simultaneously going outside of it causing incoherence.
It would be like me asking:
"if I am climbing up the walls stark raving mad, WHY do lions hunt in packs!? 😠"
The question would be 'idiotic', but still with no self-contradiction and perfectly answerable.
Oh you are making me go around and around in circles here to get an answer;
I am answering as clearly as I know how. I am not trying to obfuscate. There are however limits to the clarity one can achieve in an internet post.
OK then, what is the difference between a "truly zero probability" (which you say doesn't exist) and a "zero probability" that is not "truly" "zero probability" ?
It is the difference between 'could happen but won't' and 'couldn't happen'.
If I ask you to pick a real number at random, you could pick 5, but you wont. There is zero probability of you picking 5. But you could. You won't. If you do, I could reasonably claim that you did not pick randomly. This should not be seen as conflicting with a situation where you pick a number and then ask 'what was the probability of picking that number'?
(note also that you will pick an irrational number as the probability of picking a rational is also zero whereas the probability of picking an irrational is one : )
The probability of you picking 5+6i (a complex number) is not zero, but undefined ie you were tasked with picking a real number and a real number is what you will pick, assigning a probability to a complex number is incoherent in that scenario.
But that is exactly what I have been saying all along!
You have said that in your OP example x=0 cannot happen. It is my claim that x=0 is not part of the sample space and does not have a defined probability.
01 Feb 16
8 edits
Originally posted by humytwhitehead
...what is the difference between a "truly zero probability" (which you say doesn't exist) and a "zero probability" that is not "truly" "zero probability" ?
-you give me examples of the meaning of the latter but not the former.
Forget that question: I just thought of a much much better question than that one for you:
According to you, would the probability of an earthworm burrowing (or perhaps 'swim', rather than 'burrow', for at least part of the way ) to the exact center of the Earth be 'zero' probability?
Or do you say that even that probability is 'undefined' because there is no such thing as 'truly' zero probability?
01 Feb 16
1 edit
Originally posted by humyIs it impossible, or simply unlikely? If it is impossible then the probability is undefined. If it is possible but not probable then the probability may be zero (but not the 'true zero' )
twhitehead
Forget that question: I just thought of a much much better question than that one for you:
According to you, would the probability of an earthworm burrowing (or perhaps 'swim', rather than 'burrow', for at least part of the way ) to the exact center of the Earth be 'zero' probability?
Or do you say that even that probability is 'undefined' because there is no such thing as 'truly' zero probability?
Lets assume that such burrowing is truly impossible.
Lets think about what our 'sample space' is in the question. Suppose I say: let the sample space be all places under the earth's surface that the worm can burrow to. When you ask 'what is the probability of' then you are asking about an event. But what is an event? An event is a subset of the sample space. So when you ask 'what is the probability of the worm burrowing to the centre of the earth' you are being incoherent.
If instead you say 'let the sample space be the worm burrowing to all places under the earth' then you are violating the definition of 'sample space' as the definition requires that the events be possible.