SMp(x) Distribution
Evaluate the SMp(x) probability distribution function online. Enter six parameters to compute SMp(x) values, visualize the distribution curve, and simulate normal and Poisson distributions.
About This Calculator
The SMp(x) Distribution Calculator evaluates the versatile SMp(x) probability distribution function developed by Terman Frometa-Castillo as part of the Statistical Models project (SMp). Unlike standard probability distributions that have fixed formulas, the SMp(x) function uses six adjustable parameters to simulate virtually any known probability distribution — from the normal and Poisson to binomial and exponential distributions.
To use the calculator, enter the six model parameters: PXmin (lower limit of x), Xmax (upper limit of x), ML (the value where the function reaches its maximum), powers p₁ and p₂ (both greater than zero, controlling curve shape), and Max (the peak value of the function). Then enter an independent variable x to evaluate SMp(x). The calculator applies the piecewise formula: for x below PXmin or above Xmax the result is zero, between PXmin and ML it rises following a power law with exponent p₁, and between ML and Xmax it decays with exponent p₂.
The calculator also generates a full distribution curve across the range from PXmin to Xmax, allowing you to visualize the shape. Toggle between Curve and Distribution view modes to inspect the function from different perspectives. For the special case where p₁ = p₂ > 1 and the function is symmetric (ML at the midpoint), the SMp(x) function can serve as a normal distribution with mean μ = ML.
Methodology
The SMp(x) function is defined piecewise: SMp(x) = 0 for x < PXmin or x > Xmax; SMp(x) = [(x − PXmin)/(ML − PXmin)]^p₁ × Max for PXmin ≤ x ≤ ML; and SMp(x) = [(Xmax − x)/(Xmax − ML)]^p₂ × Max for ML ≤ x ≤ Xmax. The powers p₁ and p₂ must be greater than zero. This flexible parameterization allows a single function to represent a wide array of distribution shapes by adjusting the six parameters.
Regional Notes
Global: The SMp(x) distribution is a mathematical model developed by Terman Frometa-Castillo and is used by statisticians and researchers worldwide. Its applications span all regions equally.
Frequently Asked Questions
What is the SMp(x) distribution?
The SMp(x) distribution is a six-parameter probability function developed by Terman Frometa-Castillo. It can simulate virtually every known probability distribution, including the normal distribution, Poisson distribution, binomial distribution, and exponential distribution. The six parameters are PXmin (lower limit), Xmax (upper limit), ML (most likely value), p₁ and p₂ (shape powers), and Max (maximum value).
How do I use the SMp(x) distribution calculator?
To use the SMp(x) distribution calculator: (1) Enter the six model parameters: PXmin (lower limit), Xmax (upper limit), ML (most likely x value), p₁ and p₂ (powers greater than zero), and Max (maximum value). (2) Enter the independent variable x to evaluate the function. (3) Click Calculate to compute SMp(x) and view the distribution curve across the range from PXmin to Xmax.
What parameters does the SMp(x) function use?
The SMp(x) function uses six parameters: PXmin is the lower limit value of x, Xmax is the upper limit value of x, ML is the value of x where SMp(x) reaches its maximum, p₁ and p₂ are shape powers that are both greater than zero and determine the curve shape, and Max is the maximum value of the SMp(x) function. The function evaluates to zero for x values below PXmin or above Xmax.
Can the SMp(x) distribution simulate a normal distribution?
Yes, the SMp(x) distribution can simulate a normal distribution when: p₁ = p₂ > 1, the function is symmetric (ML = (PXmin + Xmax)/2), PXmin ≥ 0, and Max = (p₁ + 1) / [2(ML - PXmin)]. When these conditions are satisfied, the mean μ equals ML and the area under the curve equals 1, making it equivalent to a standard normal distribution.
How is the SMp(x) function calculated?
The SMp(x) function is calculated using a piecewise formula: For x < PXmin or x > Xmax, SMp(x) = 0. For PXmin ≤ x ≤ ML, SMp(x) = [(x - PXmin)/(ML - PXmin)]^p₁ × Max. For ML ≤ x ≤ Xmax, SMp(x) = [(Xmax - x)/(Xmax - ML)]^p₂ × Max. This flexible formula allows it to model a wide variety of distribution shapes by adjusting the six parameters.
What distributions can the SMp(x) function simulate?
The SMp(x) function can simulate virtually every known probability distribution by choosing appropriate parameter values. Common examples include the normal distribution (symmetric with p₁ = p₂ > 1), Poisson distribution (for discrete data with PXmin ≥ 0 and p₁, p₂ > 1), binomial distribution, and exponential distribution. The function is particularly useful for modeling custom distributions that don't fit standard forms.
What are the applications of the SMp(x) distribution?
The SMp(x) distribution has applications in statistical modeling, data analysis, probability theory, and simulation. It is used by researchers and statisticians to model complex probability distributions that don't fit standard parametric forms. The function is especially valuable in fields like risk analysis, quality control, environmental modeling, and any domain requiring flexible probability density functions.
Is the SMp(x) distribution a probability density function?
Yes, the SMp(x) function can serve as a probability density function (PDF) when the area under the curve equals 1, which is achieved by setting Max = (p₁ + 1) / [2(ML - PXmin)] for symmetric cases. For general cases, the function describes a generalized distribution shape that can be normalized. The SMp(x) model is designed to accommodate both continuous and discrete probability distributions.