Post History
#5: Post edited
by
Ooker
·
2026-05-07T08:29:07Z (4 months ago)
From [Can expectation be calculated if the events aren't numbered?](https://math.codidact.com/posts/296069), I get that as long as we can do Lebesgue integral, then you can randomly pick one and call it whatever we want, including expected value. But I think as we gradually equip more properties on the codomain of the measure so that it becomes $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), then that integral may not become the classical expectation as we know. I'd like to see how important results in classical statistics be generalized, namely expectation, variance, PDF, CDF, Gauss distribution, central limit theorem, etc. My guess is that the formulae for the expected value and the variance don't change: $\operatorname{E}[X] = \frac{\int f(x)}{∫dx}$ and $\operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 ight]$. The generalized Gauss distribution should at least be a universal point that all sequences converge at. My guess is from observing how the central limit theorem works: if you continuously applying the convolution operation on a function, then you will get the Gauss distribution at the end. See [this video of 3blue1brown](https://youtu.be/zeJD6dqJ5lo?si=hTnPb0QeZinAZwIc&t=930).- As I would like to see generalization on useful spaces, I prefer the specific spaces to be $\mathbb R$ and the space of harmonic functions, as they both play significant roles in physics. I'd like you to help me pick 2 useful superspaces of them, then gradually equip properties so at the end we have those specific spaces.
- Note that the probability to for an event to happen is still in $[0, 1]$.
- From [Can expectation be calculated if the events aren't numbered?](https://math.codidact.com/posts/296069), I get that as long as we can do Lebesgue integral, then you can randomly pick one and call it whatever we want, including expected value. But I think as we gradually equip more properties on the codomain of the measure so that it becomes $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), then that integral may not become the classical expectation as we know. I'd like to see how important results in classical statistics be generalized, namely expectation, variance, PDF, CDF, Gauss distribution, central limit theorem, etc.
- My guess is that the formulae for the expected value and the variance don't change: $\operatorname{E}[X] = \frac{\int f(x)}{∫dx}$ and $\operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 ight]$. The classical Gauss distribution stems from the classical central limit theorem: [continuously applying the convolution operation on a function, then you will get the Gauss distribution at the end](https://youtu.be/zeJD6dqJ5lo?si=hTnPb0QeZinAZwIc&t=930). So I guess the generalized central limit theorem is the Cauchy theorem, and the Gauss distribution is the universal point that all sequences converge at?
- As I would like to see generalization on useful spaces, I prefer the specific spaces to be $\mathbb R$ and the space of harmonic functions, as they both play significant roles in physics. I'd like you to help me pick 2 useful superspaces of them, then gradually equip properties so at the end we have those specific spaces.
- Note that the probability to for an event to happen is still in $[0, 1]$.
#4: Post edited
by
Ooker
·
2026-05-07T08:15:33Z (4 months ago)
From [Can expectation be calculated if the events aren't numbered?](https://math.codidact.com/posts/296069), I get that as long as we can do Lebesgue integral, then you can randomly pick one and call it whatever we want, including expected value. But I think as we gradually equip more properties on the codomain of the measure so that it becomes $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), then that integral may not become the classical expectation as we know. I'd like to see how important results in classical statistics be generalized, namely expectation, variance, PDF, CDF, Gauss distribution, central limit theorem, etc. My bet is that the expectation must contain norm, variance must contain distance, and the Gauss distribution must be a universal point for all sequence convergence.- As I would like to see generalization on useful spaces, I prefer the specific spaces to be $\mathbb R$ and the space of harmonic functions, as they both play significant roles in physics. I'd like you to help me pick 2 useful superspaces of them, then gradually equip properties so at the end we have those specific spaces.
- Note that the probability to for an event to happen is still in $[0, 1]$.
- From [Can expectation be calculated if the events aren't numbered?](https://math.codidact.com/posts/296069), I get that as long as we can do Lebesgue integral, then you can randomly pick one and call it whatever we want, including expected value. But I think as we gradually equip more properties on the codomain of the measure so that it becomes $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), then that integral may not become the classical expectation as we know. I'd like to see how important results in classical statistics be generalized, namely expectation, variance, PDF, CDF, Gauss distribution, central limit theorem, etc. My guess is that the formulae for the expected value and the variance don't change: $\operatorname{E}[X] = \frac{\int f(x)}{∫dx}$ and $\operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right]$. The generalized Gauss distribution should at least be a universal point that all sequences converge at. My guess is from observing how the central limit theorem works: if you continuously applying the convolution operation on a function, then you will get the Gauss distribution at the end. See [this video of 3blue1brown](https://youtu.be/zeJD6dqJ5lo?si=hTnPb0QeZinAZwIc&t=930).
- As I would like to see generalization on useful spaces, I prefer the specific spaces to be $\mathbb R$ and the space of harmonic functions, as they both play significant roles in physics. I'd like you to help me pick 2 useful superspaces of them, then gradually equip properties so at the end we have those specific spaces.
- Note that the probability to for an event to happen is still in $[0, 1]$.
#3: Post edited
by
Ooker
·
2026-05-04T14:02:03Z (5 months ago)
From [Can expectation be calculated if the events aren't numbered?](https://math.codidact.com/posts/296069), I get that as long as we can do Lebesgue integral, then you can call it whatever you want, including expected value. But I think as we gradually equip more properties that would make the codomain of the measure to become $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), that may not reduce to the classical expectation we know. I'd like to see how important results in classical statistics be generalized, namely expectation, variance, PDF, CDF, Gauss distribution, central limit theorem, etc. My bet is that the expectation must contain norm, variance must contain distance, and the Gauss distribution muse be a universal point of all sequence conversion.The answer can be from generalization to specification or reversed. In any case, I would like the specific spaces to be $\mathbb R$ and the space of harmonic functions, as they both play significant roles in physics. I'd like you to help me pick 2 useful generalized spaces that are supersets of them.Note that the probability to get an event is still in $[0, 1]$.
- From [Can expectation be calculated if the events aren't numbered?](https://math.codidact.com/posts/296069), I get that as long as we can do Lebesgue integral, then you can randomly pick one and call it whatever we want, including expected value. But I think as we gradually equip more properties on the codomain of the measure so that it becomes $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), then that integral may not become the classical expectation as we know. I'd like to see how important results in classical statistics be generalized, namely expectation, variance, PDF, CDF, Gauss distribution, central limit theorem, etc. My bet is that the expectation must contain norm, variance must contain distance, and the Gauss distribution must be a universal point for all sequence convergence.
- As I would like to see generalization on useful spaces, I prefer the specific spaces to be $\mathbb R$ and the space of harmonic functions, as they both play significant roles in physics. I'd like you to help me pick 2 useful superspaces of them, then gradually equip properties so at the end we have those specific spaces.
- Note that the probability to for an event to happen is still in $[0, 1]$.
#2: Post edited
by
Ooker
·
2026-05-04T08:29:49Z (5 months ago)
- From [Can expectation be calculated if the events aren't numbered?](https://math.codidact.com/posts/296069), I get that as long as we can do Lebesgue integral, then you can call it whatever you want, including expected value. But I think as we gradually equip more properties that would make the codomain of the measure to become $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), that may not reduce to the classical expectation we know. I'd like to see how important results in classical statistics be generalized, namely expectation, variance, PDF, CDF, Gauss distribution, central limit theorem, etc. My bet is that the expectation must contain norm, variance must contain distance, and the Gauss distribution muse be a universal point of all sequence conversion.
The answer can be from generalization to specification or reversed. In any case, I would like the specific spaces to be $\mathbb R$ and the space of harmonic functions, as they both play significant roles in physics. I'd like you to help me pick 2 useful generalized spaces that are supersets of them.
- From [Can expectation be calculated if the events aren't numbered?](https://math.codidact.com/posts/296069), I get that as long as we can do Lebesgue integral, then you can call it whatever you want, including expected value. But I think as we gradually equip more properties that would make the codomain of the measure to become $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), that may not reduce to the classical expectation we know. I'd like to see how important results in classical statistics be generalized, namely expectation, variance, PDF, CDF, Gauss distribution, central limit theorem, etc. My bet is that the expectation must contain norm, variance must contain distance, and the Gauss distribution muse be a universal point of all sequence conversion.
- The answer can be from generalization to specification or reversed. In any case, I would like the specific spaces to be $\mathbb R$ and the space of harmonic functions, as they both play significant roles in physics. I'd like you to help me pick 2 useful generalized spaces that are supersets of them.
- Note that the probability to get an event is still in $[0, 1]$.
#1: Initial revision
by
Ooker
·
2026-05-04T08:24:47Z (5 months ago)
How important results in classical statistics be generalized when codomain of the measure is not $\mathbb R$?
From [Can expectation be calculated if the events aren't numbered?](https://math.codidact.com/posts/296069), I get that as long as we can do Lebesgue integral, then you can call it whatever you want, including expected value. But I think as we gradually equip more properties that would make the codomain of the measure to become $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), that may not reduce to the classical expectation we know. I'd like to see how important results in classical statistics be generalized, namely expectation, variance, PDF, CDF, Gauss distribution, central limit theorem, etc. My bet is that the expectation must contain norm, variance must contain distance, and the Gauss distribution muse be a universal point of all sequence conversion.
The answer can be from generalization to specification or reversed. In any case, I would like the specific spaces to be $\mathbb R$ and the space of harmonic functions, as they both play significant roles in physics. I'd like you to help me pick 2 useful generalized spaces that are supersets of them.