question about probability density

Can it ever make sense for a probability density to ever equal infinity (more specifically, +infinity) ?

If no, because that's nonsense, how do you think its best to explain why that's nonsense?

If yes, can you give an example of a hypothetical probability distribution where for one (and I assume only one) exact value of a continuous random variable has a probability density of infinity?

I tried googling this but got nowhere.

@humy said
Can it ever make sense for a probability density to ever equal infinity (more specifically, +infinity) ?

If no, because that's nonsense, how do you think its best to explain why that's nonsense?

If yes, can you give an example of a hypothetical probability distribution where for one (and I assume only one) exact value of a continuous random variable has a probability density of infinity?

I tried googling this but got nowhere.

I can think of things like the Dirac delta "function," but that's not associated with a continuous random variable, rather a discrete one.

Probability and statistics are nowhere near my specialties, but, it seems the problem is one of constructing a cumulative distribution function F_X that has an infinite derivative at one point. The cube root function x^1/3 has an infinite derivative at 0. Work it into some piecewise-defined continuous function. Something like this, for the random variable X:

F_X(x) = 0, for x < -1
F_X(x) = 0.5 + 0.5*x^1/3, for -1 <= x < 1
F_X(x) = 1, for x >= 1

Your probability density function f_X(x) is the derivative of F_X(x), which will be +infinity at x=0.

But I don't know if I'm remembering correctly all the little bits and pieces that comprise the strict definition of a cumulative distribution function. In measure theoretic treatments I believe the derivative of F_X is allowed to not exist as a real number on a set of measure zero.

@Soothfast

Although that doesn't answer my question, I appreciate your input.

I tried but so far have failed to find any example of a conventionally defined continuous distribution where it can make sense for probability density of some specific value of x equaling +infinity.
I have also tried but so far have failed to CONTRIVE a hypothetical continuous distribution with that property.
Although the pure maths of such a distribution, ignoring all hypothetical applications, might make sense, not sure if it could ever make sense in the physical real world, which is only what I am interested in here.
This makes me intuitively suspect it probably necessarily doesn't make sense but I have been so far been unable to pin down WHY thus I could be wrong.

I have a new question about probability density;

Is it true that the conventional term "probability density" is a misnomer because it isn't ITSELF a probability at all but rather just a maths element that helps you to find a probability mass of an integral, where probability mass, NOT probability density, is the only 'true' kind of probability?
Or is it technically correct to say "probability density" really is a 'true' kind of probability?
Note that a probability density (of something) can have a value of over 1, which certainly goes against what most laypeople would intuitively think of as allowed with a 'true' probability.
I also looked up "probability" at wiki and noted it spoke of it having a value in [0, 1] interval and never over 1 albeit WITHOUT mentioning the terms "probability mass" or "probability density" so not sure if I should interpret that as necessarily meaning "probability density" isn't ITSELF a 'true' probability.

@humy said
@Soothfast

Although that doesn't answer my question, I appreciate your input.

I tried but so far have failed to find any example of a conventionally defined continuous distribution where it can make sense for probability density of some specific value of x equaling +infinity.
I have also tried but so far have failed to CONTRIVE a hypothetical continuous distribution with ...[text shortened]... ssarily doesn't make sense but I have been so far been unable to pin down WHY thus I could be wrong.

The derivative of the function F_X I constructed, which I denote by f_X, is

f_X(x) = 0, for x < -1
f_X(x) = (1/6)x^(-2/3), for -1 <= x < 1, with f_X(0)=+∞
f_X(x) = 0, for x >= 1

Integrate this from -∞ to +∞ and you get 1, as required. It is a probability density function (p.d.f.) as near as I can tell, and it gives the probability density at x=0 as being +∞. (You technically should determine f_X(0), or F'_X(0), using the limit definition of derivative to see this.) The random variable X is continuous since F_X is continuous everywhere, and f_X (i.e. F'_X) exists and is continuous everywhere except at -1, 0, and +1.

Elementary stat books might say a p.d.f. f_X has to be everywhere continuous, but in more advanced treatments this requirement is relaxed a bit, so that for instance a finite number of discontinuities may be present. Above, f_X has jump discontinuities at -1 and +1, and an infinite discontinuity at 0. I think it might be an example of what you're looking for, though I'm not sure what you mean by a "conventionally defined" continuous distribution.

In the real world something like charge density goes to infinity at the precise location of an electron (never mind the Uncertainty Principle here). But an electron is classically treated as a point in space, I believe? To compute total charge in some region containing an electron you would be integrating over the region, including the place where the electron is. Nevertheless you don't find the total charge to be infinite.

Also look here, it caught my eye:
https://tinyurl.com/y5kmnpas

@humy said
I have a new question about probability density;

Is it true that the conventional term "probability density" is a misnomer because it isn't ITSELF a probability at all but rather just a maths element that helps you to find a probability mass of an integral, where probability mass, NOT probability density, is the only 'true' kind of probability?
Or is it technically correct ...[text shortened]... hould interpret that as necessarily meaning "probability density" isn't ITSELF a 'true' probability.

Probability density is just probability per unit volume (or area or length), just as mass density is mass per unit volume. I don't see it as a misnomer.

@soothfast said
Probability density is just probability per unit volume (or area or length), just as mass density is mass per unit volume. I don't see it as a misnomer.

So you say the said probability density is probability mass per unit volume, right?
I'm not arguing about your definition (of probability density) as I know what you mean by it but saying something x (x='probability density', in this case) is y per z (y='probability mass' and z='unit volume', in this case) doesn't logically imply x goes into the same more generic category ('probability', in this case) as y, right?

@soothfast said

I'm not sure what you mean by a "conventionally defined" continuous distribution.

What I meant was a continuous distribution that has been mathematically formally described and studied. I guess if there is a wiki page for a distribution then it must be. I have discovered and defined and studied some totally new distributions that I will put in my book and I would say they are not "conventionally defined" at the present time because they are currently only known about my me.

I read you post and I think you have in effect answered my question; I now think it CAN make 'sense' for a probability density to be +infinity, at least in the sense there is no self-contradiction or logical paradox in there existing a probability density of +infinity in the real world, even if no such thing actually exist in the real world.
Whether any continuous distribution which has for one of its x values a probability density of +infinity actually applies to the real world is a different matter and I am not sure if any does. But, even if no such distribution, that doesn't imply it makes no sense for there to be one.

Thanks.

@humy said
What I meant was a continuous distribution that has been mathematically formally described and studied. I guess if there is a wiki page for a distribution then it must be. I have discovered and defined and studied some totally new distributions that I will put in my book and I would say they are not "conventionally defined" at the present time because they are currently only kno ...[text shortened]... even if no such distribution, that doesn't imply it makes no sense for there to be one.

Thanks.

At https://tinyurl.com/y665z8wq the equation (6.125) is a practical example of a p.d.f. with infinite probability density at a couple points. Something to do with nonlinear dynamics.

EDIT: I'm unsure of what subtext might be lurking behind (6.125), because a glance at its graph shows the area under the curve to be easily greater than 1. Hmm...