B Continuous random variable: Zero probablity

[FactChecker]

Suppose I throw a dart at an (X,Y) grid [0,1]x[0,1], and say that the result is the exact x-coordinate of the dart's center of mass. Then I have done an experiment in the physical world where the result has an infinite accuracy. In the physical world, we will not be able to determine the exact result with infinite accuracy, but nevertheless, it has happened. That exact result had zero probability. (Here I am assuming that there is no quantization in nature of the space-time coordinate system.) The fact that we can not determine and record the result with infinite accuracy is not relevant.

 

[PeroK]

Suppose I throw a dart at an (X,Y) grid [0,1]x[0,1], and say that the result is the exact x-coordinate of the dart's center of mass. Then I have done an experiment in the physical world where the result has an infinite accuracy. In the physical world, we will not be able to determine the exact result with infinite accuracy, but nevertheless, it has happened. That exact result had zero probability. (Here I am assuming that there is no quantization in nature of the space-time coordinate system.) The fact that we can not determine and record the result with infinite accuracy is not relevant.

This assumes that the dart has a well-defined centre of mass and that the centre of mass has a well-defined position without being measured. Mathematically, this is fine. But, I don't think these mathematical concepts extend to the real world with "infinite accuracy". Whether you consider Quantum Mechanics or not. Classically, the atoms in a dart are never at rest, so its centre of mass is never at rest either.

To me, this is a fundamental difference between mathematics and reality. An experiment cannot generate these mathematical things, like an x-coordinate to infnite accuracy.

 

[jbriggs444]

If you work through the math, as n grows large, you actually get real roots with probability one (i.e. the probability of complex root is zero)

Please share this working through.

Suppose that we select a, b and c from the uniform distribution over [-1,+1]. We get a non-zero probability of the result having a discriminant $b^2-4ac$ greater than zero and a non-zero probability of the discriminant having a value less than zero.

Now suppose instead that we select A, B and C from a uniform distribution over [-n,+n]. This is equivalent to selecting a, b and c as above from a uniform distribution over [-1,+1] and setting A=na, B=nb, C=nc. The resulting discriminant is then given by $(nb)^2-4(na)(nc) = n^2(b^2-4ac)$. The distribution of signs of the discriminant is independent of n.

The limiting probability of a complex root is not zero.

 

[FactChecker]

This assumes that the dart has a well-defined centre of mass and that the centre of mass has a well-defined position without being measured. Mathematically, this is fine. But, I don't think these mathematical concepts extend to the real world with "infinite accuracy". Whether you consider Quantum Mechanics or not. Classically, the atoms in a dart are never at rest, so its centre of mass is never at rest either.

To me, this is a fundamental difference between mathematics and reality. An experiment cannot generate these mathematical things, like an x-coordinate to infnite accuracy.

It doesn't matter if there is an atom there or not or if atoms are moving. It doesn't matter how the center of mass is defined. However it is defined, If that location exists with infinite precision in the time-space coordinate system, then the experiment has a result that has infinite precision. The probability of the infinite precision result is zero. We can never measure or record it with infinite precision, but that is a different issue.

 

[PeroK]

It doesn't matter if there is an atom there or not or if atoms are moving. It doesn't matter how the center of mass is defined. However it is defined, If that location exists with infinite precision in the time-space coordinate system, then the experiment has a result that has infinite precision. The probability of the infinite precision result is zero. We can never measure or record it with infinite precision, but that is a different issue.

The task, I believe, is to choose a random number. Your answer is to throw a dart at a board and take the "x-coordinate". The issue is whether choosing a random number includes identifying it.

Since you can't actually identify the one you have chosen, does that count as choosing one?

In the finite case it wouldn't work. If you have to draw the first round of a tennis tournament, you have to actually produce the players' names. You can't have match 1 between two unknown players who have have been chosen by a process that didn't actually identify them!

It would be different if you identified it by some property, like the solution to a transendental equation. That still identifies the number uniquely.

But, to say that your random number is "the x-coordinate of that dart (whatever it is)" doesn't feel like you've actually chosen a specific number.

 

[FactChecker]

The task, I believe, is to choose a random number. Your answer is to throw a dart at a board and take the "x-coordinate". The issue is whether choosing a random number includes identifying it.

Since you can't actually identify the one you have chosen, does that count as choosing one?

The question is whether there are always physical-world constraints on the accuracy of experimental results. I believe that there are examples where the only limitation is on our ability to observe and record the result with infinite accuracy, not on the result itself. I think that the dart-throw is a physical experiment where the result had a zero prior probability. The dart landed. The x-coordinate position exists (unless time-space coordinates are quantized). The physical result occurred. The issue of whether I was able to observe it and record it with infinite precision is a separate issue.

 

[PeroK]

The question is whether there are always physical-world constraints on the accuracy of experimental results. I believe that there are examples where the only limitation is on our ability to observe and record the result with infinite accuracy, not on the result itself. I think that the dart-throw is a physical experiment where the result had a zero prior probability. The dart landed. The x-coordinate position exists (unless time-space coordinates are quantized). The physical result occurred. The issue of whether I was able to observe it and record it with infinite precision is a separate issue.

I've had an idea that I'll post on another thread.

 

[StoneTemplePython]

I agree. My point is that without information about the physical process, the true fact is that our knowledge has not improved and the probabilities are unchanged. For all we know, the method used to select the random number have itself been randomly selected out of a multitude of methods. I believe it is legitimate to talk about a continuous CDF with zero probability at every single point, even for a physical, non-theoretical case. Whether that is physically possible seems to be a "religious" issue.

I like this, though I trust you mean PDF, not CDF. (If you actually mean CDF... well then we have a big problem as it seems to mean your PDF doesn't integrate to one and so on... if you are mapping all probability to $\infty$ the random variable would seem to be pathologically defective).

To a large degree, a lot of this discussion reminds of when I tell someone the convention in mathematics that

$1.99999999999... = 2$

It is a convention and if someone says they don't accept it, well fine, but this creates all kinds of contradictions with basic rules of arithmetic and requires a lot of special, new inventions. You point that out to them and after a bit of thought they either accept the convention, or they respond by saying that $1.99999999999...$ doesn't exist. But existence has nothing to do with it. And even if people don't like the beauty of math, I think people get that using things like infinite series and continuity can be immensely useful.

 

[Stephen Tashi]

Suppose I throw a dart at an (X,Y) grid [0,1]x[0,1], and say that the result is the exact x-coordinate of the dart's center of mass. Then I have done an experiment in the physical world where the result has an infinite accuracy.

We can suppose such a thing can happen, but If we suppose that you (or Nature) can pick an exact mathematical point from a continuous distribution then we have made an assumption about physics.

In the physical world, we will not be able to determine the exact result with infinite accuracy, but nevertheless, it has happened. That exact result had zero probability. (Here I am assuming that there is no quantization in nature of the space-time coordinate system.) The fact that we can not determine and record the result with infinite accuracy is not relevant.

I agree that the following physical situations are different:

1) Nature cannot select an exact result from a continuous probability distribution.

2) Nature can select an exact result from a continuous probability distribution, but we cannot measure what nature has done exactly.

So the fact we cannot measure an exact result from an experiment doesn't tell us whether 1) or 2) is the case.

My point about the mathematical theory of probability is that it does not assert we can do such an experiment with a dart. - i.e. it does not assert that 2) is the case.

 

[StoneTemplePython]

Please share this working through.

Suppose that we select a, b and c from the uniform distribution over [-1,+1]. We get a non-zero probability of the result having a discriminant $b^2-4ac$ greater than zero and a non-zero probability of the discriminant having a value less than zero.

Now suppose instead that we select A, B and C from a uniform distribution over [-n,+n]. This is equivalent to selecting a, b and c as above from a uniform distribution over [-1,+1] and setting A=na, B=nb, C=nc. The resulting discriminant is then given by $(nb)^2-4(na)(nc) = n^2(b^2-4ac)$. The distribution of signs of the discriminant is independent of n.

The limiting probability of a complex root is not zero.

First, carefully look at the original problem. Or as I noted in a follow-up post:

For avoidance of doubt, there is no $a$ term in $x^2 + 2bx + c = 0$ (or put differently, $a$ is fixed at one).

Now draw a graph with a square from $[-n, n]$ on the X and Y axis. Area of the square is $2n * 2n = 4n^2$. Now draw parabola associated with $b^2 -c \lt 0$. Color blue: all the area 'inside' the parabola (and bounded above by the square's top edge). For $n \geq 1$ the blue area should be $\frac{4}{3}n^{\frac{3}{2}}$. What is the ratio of blue area to total square? $\frac{ \frac{4}{3}n^{\frac{3}{2}}}{4n^2}$, or as I'd call it $O\big(\frac{1}{\sqrt n}\big)$.

 

[FactChecker]

I trust you mean PDF, not CDF.

Good point. Thanks. I worded my statement badly. I meant that the CDF is continuous, implying that the probability of any single exact resulting value is zero. -- Not that the cumulative probability is zero.

 

[FactChecker]

We can suppose such a thing can happen, but If we suppose that you (or Nature) can pick an exact mathematical point from a continuous distribution then we have made an assumption about physics.

Good point. In fact, I may have seen somewhere that in quantum theory time is in fact quantized, so location on an X-axis may also be quantized. I don't know enough to comment more than that. Even if that is true, I think that I would accept the approximation of the discrete physics with a continuous model for the purpose of ignoring any quantization of time-space.

I agree that the following physical situations are different:

1) Nature cannot select an exact result from a continuous probability distribution.

2) Nature can select an exact result from a continuous probability distribution, but we cannot measure what nature has done exactly.

So the fact we cannot measure an exact result from an experiment doesn't tell us whether 1) or 2) is the case.

My point about the mathematical theory of probability is that it does not assert we can do such an experiment with a dart. - i.e. it does not assert that 2) is the case.

I have to agree. At the finest level of detail, we may never know the answer. I will have to resign myself to the realization that, at the quantum level, the continuous CDF may not be possible. It may be an approximation.

 

[Joseph M. Zias]

Consider a block of wood whose linear density you know, say 100 g per cm. You acquire mass by spanning a distance. As that distance gets smaller so does the mass acquired, a span the thickness of a thin paper would be very small. In the limit as the span approaches zero you would of course have zero mass. The same effect is seen in spectrum analysis. As the bandwidth gets narrower the energy measured gets less, if you had a bandwidth of zero - a single frequency - you would have zero energy.

 

[sysprog]

Calling it zero probability is linguistically misleading -- only the impossible has zero probability -- calling something possible is the same as saying it has more than zero probability. The probability that 1 = 2 is 0. The probability that a number to be chosen from all real numbers will be 1.2 is > 0.