User:PolyDistCn/sandbox

Introduction

In many scientific fields, particularly in the context of Monte Carlo simulations, the Gaussian or normal distribution is commonly used to generate data samples. One typical scenario arises when the mean and variance of a parameter or population were determined by statistical methods, and the normal distribution, parameterized by these statistical values, is used in Monte Carlo simulations for modeling this behavior.

However, the normal distribution is unbounded; that is, its tails extend until positive and negative infinity. In some cases, this conflicts with the modeled behavior, e.g., for some parameters in physics, where negative values are disallowed. Therefore, to address this issue, bounded distributions are required.

The following section presents such a bounded, bell-shaped distribution, which can be regarded as a good substitute for the normal distribution in scenarios like the one mentioned before.

Mathematical formulation and details

This EFM article introduces a parameterized univariate probabilistic distribution based on polynomials. The distribution is symmetric and bounded, with bounding interval ${\textstyle [0,1]}$ (basic distribution). By shifting and scaling, the bounding interval can be adapted to any required interval ${\textstyle [a,b]}$ (generalized distribution). The parameter ${\textstyle n\geq 1}$ controls the smoothness of the distribution at its boundaries, with higher values resulting in increased smoothness.

The probability density function (PDF) is based on polynomials and defined by

$$\operatorname {PDF} _{p_{n}}(x)={\begin{cases}0&{\text{if}}\ x<0,\\p'_{n}(x)&{\text{if}}\ x\in [0,1],\\0&{\text{if}}\ 1<x,\end{cases}}$$

with

$$p'_{n}(x)=a_{n}[x(1-x)]^{n}=a_{n}\sum \limits _{k=0}^{n}{\binom {n}{k}}{(-1)^{k}}x^{n+k}.$$

The parameter ${\textstyle a_{n}}$ is proven to be

$$a_{n}={(2n+1){\binom {2n}{n}}},$$

that is, all coefficients of the polynomial ${\textstyle p'_{n}(x)}$ are integer values.

The cumulative distribution function is obtained by integration

$$\operatorname {CDF} _{p_{n}}(x)={\begin{cases}0&{\text{if}}\ x<0,\\p_{n}(x)&{\text{if}}\ x\in [0,1],\\1&{\text{if}}\ 1<x,\end{cases}}$$

with

$$p_{n}(x)=\int \limits _{0}^{x}{p'_{n}(x)}{dx}=a_{n}\sum \limits _{k=0}^{n}{\frac {{\binom {n}{k}}{(-1)^{k}}x^{n+k+1}}{n+k+1}}.$$

The following equations show the distribution functions of the basic distribution with bounding interval ${\textstyle [0,1]}$, as well as the variance (the mean of the symmetric distribution is ${\textstyle \mu =0.5}$ in all cases) for the parameter set ${\textstyle 1\leq n\leq 5}$.

- ${\textstyle n=1}$

$${\begin{aligned}a_{1}=&6\\p'_{1}(x)=&-6x^{2}+6x\\p_{1}(x)=&-2x^{3}+3x^{2}\\\sigma _{1}^{2}=&0.05\end{aligned}}$$

- ${\textstyle n=2}$

$${\begin{aligned}a_{2}=&30\\p'_{2}(x)=&30x^{4}-60x^{3}+30x^{2}\\p_{2}(x)=&6x^{5}-15x^{4}+10x^{3}\\\sigma _{2}^{2}\approx &0.036\end{aligned}}$$

- ${\textstyle n=3}$

$${\begin{aligned}a_{3}=&140\\p'_{3}(x)=&-140x^{6}+420x^{5}-420x^{4}+140x^{3}\\p_{3}(x)=&-20x^{7}+70x^{6}-84x^{5}+35x^{4}\\\sigma _{3}^{2}\approx &0.028\end{aligned}}$$

- ${\textstyle n=4}$

$${\begin{aligned}a_{4}=&630\\p'_{4}(x)=&630x^{8}-2520x^{7}+3780x^{6}-2520x^{5}+630x^{4}\\p_{4}(x)=&70x^{9}-315x^{8}+540x^{7}-420x^{6}+126x^{5}\\\sigma _{4}^{2}\approx &0.023\end{aligned}}$$

- ${\textstyle n=5}$

$${\begin{aligned}a_{5}=&2772\\p'_{5}(x)=&-2772x^{10}+13860x^{9}-27720x^{8}+27720x^{7}-13860x^{6}+2772x^{5}\\p_{5}(x)=&-252x^{11}+1386x^{10}-3080x^{9}+3465x^{8}-1980x^{7}+462x^{6}\\\sigma _{5}^{2}\approx &0.019\end{aligned}}$$