SMp(x) Distribution Calculator

At its core, the calculator evaluates the SMp function based on a set of defined parameters including the lower boundary (PXmin), upper boundary (Xmax), most likely value (ML), and distribution shape parameters (p1 and p2). The primary output, SMp, represents the value of the distribution function evaluated at a given variable x. Probabilities are derived by integrating or computing the cumulative distribution functions over designated intervals: finding the probability above a threshold x1, below a threshold x2, or strictly between two boundaries x3 and x4. Mathematically, these relations translate to evaluating specific distribution functions configured through your Maximum, ML, and PXmin inputs.

Imagine you are analyzing a dataset where the most likely value (ML) is 50, the minimum possible value (PXmin) is 10, and the absolute maximum value (Xmax) is 100. Assume your shape parameters p1 and p2 are set to 2 and 1.5 respectively, with a Maximum normalization limit of 1. If you want to find the probability of observing a value below x = 40, the calculator applies the distribution function using these parameters to integrate the lower tail. By feeding these concrete figures into the algorithm, the tool computes the cumulative density up to 40, yielding a precise probability output—for instance, a cumulative probability of 0.345, indicating a 34.5% chance of encountering a value under 40 in this distribution.

To ensure the most accurate results from your distribution analysis, always double-check that your boundary conditions—such as PXmin and Xmax—properly encompass your target variable x. A common pitfall is entering parameters where x falls outside the theoretical minimum and maximum bounds, which can result in undefined mathematical states or zero probabilities. Additionally, verify your scale and shape parameters (p1, p2) against your raw data source before running comparative probability-between queries.