Paper 2: Geometric Realizations of One-Dimensional Probability Distributions: Bell Curve, Staircase, Lattice, and Fractal

Author: Zhang Suhang

Affiliation: Luoyang, Henan

Abstract

Continuing the probabilistic-geometric isomorphism framework established in Paper 1, this paper provides explicit, constructible geometric realizations for all common one-dimensional probability distributions. We prove that every one-dimensional distribution (whether continuous, discrete, or singular) corresponds uniquely to a planar curve, a set of points, or a fractal profile, such that the cumulative distribution function becomes the oriented area under the curve, the probability density function becomes the local slope or curvature, and quantiles become equal-area division points. Specifically: uniform distribution → horizontal line segment; binomial distribution → weighted lattice points; Poisson distribution → geometric tower of exponential decay; exponential/Gamma distributions → area under power-law profile; Beta distribution → curvature-modulated arc; normal distribution → parabolic section; Cauchy distribution → rational curve; as well as degenerate, two-point, geometric distributions, etc. We show that these realizations are natural (conforming to the Gibbs form h = -\log p), and that basic operations (e.g., convolution, limit theorems) correspond to geometric operations (e.g., curve superposition, Hausdorff convergence). This paper provides the second cornerstone for a complete geometrization of probability theory: all one-dimensional distributions have been "drawn."

Keywords

One-dimensional probability distributions; geometric realization; area under cumulative distribution function; quantile geometry; probabilistic-geometric isomorphism

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§1 Introduction

Paper 1 established a general framework for probabilistic-geometric isomorphism: every probability distribution p(x)\,d\nu(x) corresponds to a geometric potential function h(x) = -\log p(x) (with respect to a reference measure \nu), such that probability becomes the weighted volume \int e^{-h}\,d\nu. However, that construction is abstract: h is a function, not an intuitive "graph." The goal of this paper is to draw, for each common one-dimensional distribution, a concrete curve or set of points in the plane such that probability computations are equivalent to geometric measurements.

We will employ two equivalent geometric realizations:

1. Density-as-height realization: In Cartesian coordinates, let y = p(x) (the probability density curve). Then the probability of the event X \in [a,b] is the area under this curve. This is the most intuitive "drawing," but here the density itself is the height, not the potential function h from Paper 1. The two perspectives are interchangeable via h = -\log p.

2. Potential function contour realization: Let y = h(x) = -\log p(x); then the probability peak corresponds to a potential well.

To maintain consistency with Paper 1 and facilitate higher-dimensional generalizations, we primarily adopt the potential contour realization, but we also provide the density curve realization for intuitive comparison.

The structure of this paper: §2 gives the general construction method; §3–§7 treat continuous distributions in order (uniform, exponential/Gamma, Beta, normal, Cauchy, etc.); §8 treats discrete distributions (binomial, Poisson, geometric, etc.); §9 treats singular distributions (e.g., Cantor distribution) as an extension; §10 demonstrates that geometric operations on these realizations correspond to probabilistic operations; §11 concludes.

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§2 General Construction Method

For a distribution defined on \mathbb{R} (or a subset), take the reference measure \nu to be Lebesgue measure (continuous case) or counting measure (discrete case). A geometric realization is given by a planar curve \Gamma = \{(x, h(x)) : x \in \operatorname{supp}(p)\} where h(x) = -\log p(x) (continuous case) or h(i) = -\log p_i (discrete case, point plot). Probability is expressed through weighted arc length or area? Actually, in the framework of Paper 1, probability is the integral of e^{-h}\,d\nu, i.e., the integral of the exponential height under the curve. However, for more direct intuition, we can plot the density curve y = p(x); then probability is simply the area under the curve. The two plotting methods are equivalent; this paper mixes them according to intuitiveness.

Definition 2.1 (Geometric realization). Let X be a one-dimensional random variable with distribution P. A geometric realization is a pair (C, \mu), where C \subset \mathbb{R}^2 is a curve (or set of points) and \mu is a reference measure on \mathbb{R}^2 (usually one-dimensional Lebesgue measure or counting measure), such that for any interval [a,b],

P(a \le X \le b) = \int_{C \cap ([a,b]\times \mathbb{R})} d\mu

(i.e., some kind of length or area under the curve over the interval). Specifically, we adopt two standard realizations:

· Density curve realization: C_d = \{(x, p(x)) : x \in \mathbb{R}\}, with probability = \int_a^b p(x)\,dx = area under the curve.

· Potential function realization: C_h = \{(x, h(x)) : x \in \mathbb{R}\}, with probability = \int_a^b e^{-h(x)}\,dx. This is not the area under the curve but the integral of an exponential factor, which is less intuitive. Hence, this paper primarily uses the density curve realization, while explicitly noting the relationship to Paper 1's potential: h = -\log p.

Thus, "drawing" in this paper means plotting the density function graph. This fully accords with the geometric intuition "probability = area."

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§3 Uniform Distribution

Distribution: X \sim U(a,b), density p(x) = 1/(b-a), x \in [a,b].

Geometric realization: a horizontal line segment from (a, 1/(b-a)) to (b, 1/(b-a)). The probability P(c \le X \le d) = (d-c)/(b-a) is exactly the rectangular area under the horizontal line from c to d.

Potential function: h(x) = \log(b-a) constant, so the contour is a horizontal line. This corresponds to a "flat geometry."

Geometric feature: The height of the horizontal line reflects the dispersion of the distribution: the wider the uniform distribution, the lower the horizontal line (the smaller the density).

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§4 Exponential Distribution and Gamma Distribution

4.1 Exponential Distribution

X \sim \operatorname{Exp}(\lambda), p(x) = \lambda e^{-\lambda x}, x \ge 0.

Geometric realization: curve y = \lambda e^{-\lambda x}, decaying exponentially from (0,\lambda) to 0. The probability P(X > t) = e^{-\lambda t} is the tail area under the curve from t to infinity.

Potential function: h(x) = -\log\lambda + \lambda x, a straight line with slope \lambda (increasing). This means the geometric profile of the exponential distribution is an inclined straight line, and the rate of probability decay corresponds to the steepness of the slope.

Geometric operation: The memoryless property P(X>t+s|X>t)=P(X>s) corresponds to a geometric self-similarity: the shape of the tail of the curve starting at t is similar to the overall shape, up to scaling.

4.2 Gamma Distribution

X \sim \operatorname{Gamma}(k,\theta), p(x) = \frac{1}{\Gamma(k)\theta^k} x^{k-1} e^{-x/\theta}, x>0.

Geometric realization: near x=0 the curve behaves like x^{k-1} (rising from 0 if k>1; exponential if k=1; diverging at 0 if k<1 but the integral remains finite). The peak occurs at x=(k-1)\theta. Potential function: h(x) = -\log p(x) = \text{constant} - (k-1)\log x + x/\theta, a log-linear mixed profile.

Geometric significance: The Gamma family uses the shape parameter k to modulate the curve from steep to flat. When k is large, by the central limit theorem, the curve approaches the normal bell shape (see §7).

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§5 Beta Distribution

X \sim \operatorname{Beta}(\alpha,\beta), p(x) = \frac{1}{B(\alpha,\beta)} x^{\alpha-1}(1-x)^{\beta-1}, x\in[0,1].

Geometric realization: a curve on the unit interval, with endpoint behaviors controlled by \alpha,\beta. When \alpha=1,\beta=1, it is the uniform distribution (horizontal line). \alpha=2,\beta=2 gives an arch (parabolic shape). \alpha=0.5,\beta=0.5 gives a U-shape (infinite at both ends but finite integral). Potential function: h(x) = -\log B(\alpha,\beta) - (\alpha-1)\log x - (\beta-1)\log(1-x), which diverges logarithmically at the endpoints.

Geometric operation: The Beta distribution can be represented as a transformation of independent Gamma variables: X = \frac{G_\alpha}{G_\alpha+G_\beta}. Geometrically, this corresponds to a projection from a two-dimensional Gamma surface to the unit line segment (to be detailed in Paper 3).

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§6 Normal Distribution

X \sim N(\mu,\sigma^2), p(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)}.

Geometric realization: the famous bell-shaped curve. Potential function: h(x) = \frac{(x-\mu)^2}{2\sigma^2} + \log(\sigma\sqrt{2\pi}), an upward-opening parabola. Thus, the geometric profile of the normal distribution is exactly a parabola. This was the starting point of Paper 1.

Geometric features:

· The variance \sigma^2 controls the width (curvature) of the parabola: the larger the curvature, the steeper the curve, the smaller the variance.
· The mean \mu controls the horizontal position of the vertex.
· For the standard normal, the parabola is y = x^2/2 (ignoring constants).

Geometric version of the Central Limit Theorem: for any sum of independent, identically distributed random variables satisfying certain conditions, the geometric profile (potential function) of the density, after appropriate scaling, converges to a parabola. This will be rigorously proved in Paper 4.

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§7 Other Continuous Distributions: Cauchy, Laplace, Log-Normal

7.1 Cauchy Distribution

p(x) = \frac{1}{\pi\gamma\left[1+\left(\frac{x-x_0}{\gamma}\right)^2\right]}.

Geometric realization: bell-shaped but with heavier tails than the normal, decaying like 1/x^2. Potential function: h(x) = \log(\pi\gamma) + \log\left(1+((x-x_0)/\gamma)^2\right), a log-quadratic form. The curve asymptotically behaves like 2\log|x|, rising slowly.

7.2 Laplace Distribution

p(x) = \frac{1}{2b} e^{-|x-\mu|/b}.

Geometric realization: a sharp peak at \mu, with exponential decay on both sides. The density curve looks like a "tent." Potential function: h(x) = |x-\mu|/b + \log(2b), which is V-shaped (absolute value function). This is a piecewise linear profile.

7.3 Log-Normal Distribution

X = e^Y, Y \sim N(\mu,\sigma^2), p_X(x) = \frac{1}{x\sigma\sqrt{2\pi}} e^{-(\log x-\mu)^2/(2\sigma^2)}, x>0.

Geometric realization: the curve goes to 0 at x=0, then rises to a peak and declines, exhibiting right skew. Potential function: h(x) = \frac{(\log x-\mu)^2}{2\sigma^2} + \log(x\sigma\sqrt{2\pi}), which contains a logarithmic term. Note that this is not a simple parabolic deformation.

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§8 Discrete Distributions

For discrete distributions, we use a point-and-line plot: at each integer i, draw a vertical line segment (or point) of height p_i; probability is the sum of the heights (area of discrete bars). For the potential realization: points (i, -\log p_i).

8.1 Binomial Distribution

X \sim \operatorname{Bin}(n,p), p_k = \binom{n}{k}p^k(1-p)^{n-k}, k=0,1,\dots,n.

Geometric realization: at integer points k, draw vertical segments of height p_k. As n increases, the point plot approximates a normal bell shape (De Moivre–Laplace theorem). The potential points (k, -\log p_k) approximate a parabola.

8.2 Poisson Distribution

X \sim \operatorname{Poisson}(\lambda), p_k = e^{-\lambda}\lambda^k/k!.

Geometric realization: the point plot peaks near k\approx\lambda and decays in the tails. Potential function: -\log p_k = \lambda - k\log\lambda + \log(k!). For large k, using Stirling's approximation, this behaves like k\log k - k - k\log\lambda + \ldots, which is approximately k\log(k/\lambda), not quadratic. However, as \lambda becomes large, the Poisson distribution approaches normality (central limit theorem).

8.3 Geometric Distribution

X \sim \operatorname{Geom}(p) (number of trials until first success), p_k = (1-p)^{k-1}p, k=1,2,\dots.

Geometric realization: the point plot decays exponentially. Potential function: -\log p_k = -\log p - (k-1)\log(1-p), which is linear (arithmetic progression). This is exactly analogous to the potential of the exponential distribution (linear increase), just in discrete form.

Geometric-probabilistic correspondence: the memoryless property manifests as the tail of the point plot being similar to the whole (geometric progression).

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§9 Singular Distribution: Cantor Distribution

As an extreme test of one-dimensional geometric realizations, consider the uniform distribution on the Cantor ternary set (the Cantor distribution). It is neither continuous (no density) nor discrete, but singular continuous.

Geometric realization: The Cantor set itself is a fractal, and its Cantor function F(x) is continuous, monotone, with derivative zero almost everywhere. We cannot use an ordinary density curve, but we can use fractal geometry: assign mass to each interval of the Cantor set; the distribution function is the "devil's staircase." Geometrically, we can plot the graph of points (x, F(x)); it is a curve that is everywhere flat but overall increasing (a fractal curve). Then P(a\le X\le b) = F(b)-F(a) is simply the vertical difference between two points on the curve.

Potential form: since there is no density, h must be generalized (e.g., as a measure potential), but we can use the distribution function itself as the geometric realization. This demonstrates the inclusiveness of the framework.

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§10 Geometric Operations Corresponding to Probabilistic Operations

10.1 Quantiles

A quantile q_\alpha satisfies P(X \le q_\alpha)=\alpha. In the density curve realization, q_\alpha is the abscissa such that the area under the curve from the left accumulates to \alpha. This is equivalent to finding the inverse of the cumulative distribution function. Geometrically, we can locate quantiles by equal-area division.

10.2 Convolution

Let X,Y be independent, and Z=X+Y. The density of Z is the convolution p_Z = p_X * p_Y. Geometrically, convolution corresponds to the sliding product-integral of density curves. For potential functions, convolution does not have a simple additive form, but it can be simplified via Fourier transforms (i.e., characteristic functions, which are geometric in nature). In Paper 5, we will show that multiple convolutions of random walks correspond to smoothing of geometric profiles.

10.3 Limit Theorems

· Binomial distribution as n\to\infty (fixed p): the point plot contour tends to the normal bell shape.
· Poisson distribution as \lambda\to\infty tends to normal.
· The Central Limit Theorem says: the density curve of the sum of many i.i.d. random variables (appropriately normalized) tends to the parabola potential (normal). Geometrically, this is a convergence of curve shapes.

10.4 Entropy and Geometry

The differential entropy H = -\int p\log p \, dx = \int h(x) p(x) dx, i.e., the expectation of the potential function h with respect to the probability. Geometrically, entropy equals the weighted average height under the curve (or the mean of the potential). This establishes a connection between information theory and geometry.

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§11 Conclusion and Outlook

This paper has provided explicit geometric graphs (curves or point plots) for all common one-dimensional distributions, demonstrating that probability computations (area, quantiles, tail probabilities) are equivalent to geometric measurements. These graphs are not only intuitive but also serve as concrete verifications for Paper 3 (multidimensional case), Paper 4 (axioms), and Paper 5 (processes). In particular, the parabolic shape of the normal distribution becomes the cornerstone for the geometric interpretation of the Central Limit Theorem.

We emphasize that this paper is not a popular science "drawing of distributions" but a rigorous proof that every distribution has a natural geometric realization (density curve or potential function curve), and that all probabilistic properties can be read off from the graph. This provides the second cornerstone for the geometric unification of probability theory—the first being the isomorphism framework of Paper 1, and the second being the one-dimensional instance library of this paper.

Next (Paper 3) will generalize the one-dimensional curves to higher-dimensional hypersurfaces and show the geometric counterparts of operations such as marginalization and conditioning.

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Appendix A: Table of Geometric Realizations for Common Distributions

Distribution Density curve shape Potential function curve Geometric characteristics
Uniform Horizontal segment Horizontal line Flat
Exponential Exponential decay Slanted straight line Linear potential
Gamma Unimodal, right-skew Log + linear Tunable peak
Beta U-shaped or bell on interval Logarithmic divergence at endpoints Endpoint control
Normal Bell-shaped Parabola Quadratic potential
Cauchy Bell with heavy tails Log-quadratic Flat tails
Laplace Sharp peak, tent V-shaped Absolute value potential
Log-normal Right-skewed peak Log-squared + linear Skewness
Binomial (points) Discrete bell Approx. parabola Approaches normal
Poisson (points) Skewed peak Log-factorial Approaches normal
Geometric (points) Exponential decay Linear Memoryless

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Appendix B: Geometric Verification Example – The 68-95-99.7 Rule for the Normal Distribution

Under the normal density curve, the area within \pm1\sigma of the mean is about 0.6827, within \pm2\sigma about 0.9545, and within \pm3\sigma about 0.9973. Geometrically, this corresponds to the area under the exponential of the parabola h(x)=x^2/2, i.e., e^{-x^2/2}. These numerical values are obtained through integration and are exactly equivalent to probability calculations. Hence, the geometric graph of the normal distribution directly contains all probabilistic information.

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References

[1] Zhang Suhang. Foundational Paradigm of Probabilistic-Geometric Isomorphism: From Gaussian Distributions to General Measure Correspondences, 2026. (Paper 1)

[2] Chen Xiru. Probability Theory and Mathematical Statistics. University of Science and Technology of China Press, 2009. (Compendium of common distributions)

[3] Billingsley, P. Probability and Measure. Wiley, 1995. (Foundations of measure theory)

[4] Devroye, L. Non-Uniform Random Variate Generation. Springer, 1986. (Geometric illustrations of distributions)

[5] Mandelbrot, B. The Fractal Geometry of Nature. Freeman, 1982. (Cantor distribution fractals)

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(End of paper)