SMp Distribution Calculator

What the SMp(x) Model Is

SMp(x) is a flexible six-parameter function that can mimic many continuous and discrete distribution shapes. It uses lower limit PXmin, upper limit Xmax, mode location ML, powers p1 and p2, and peak Max. Related tools: Normal Distribution Calculator and Poisson Distribution Calculator.

Piecewise Definition

For PXmin ≤ x ≤ ML:

$$\mathrm{SMp}(x) = \left(\frac{x - PXmin}{ML - PXmin}\right)^{p_1} \times Max$$

For ML ≤ x ≤ Xmax:

$$\mathrm{SMp}(x) = \left(\frac{Xmax - x}{Xmax - ML}\right)^{p_2} \times Max$$

Outside the interval the function is zero.

Probability Queries

This tool can estimate tail and interval probabilities by numerically integrating SMp(x) over the domain. That is useful when the curve is treated as an unnormalized density model.

Example

With PXmin = 0, Xmax = 100, ML = 50, p1 = p2 = 2, and Max = 0.04, the curve peaks at x = 50 and falls smoothly toward both limits.