SMp(x) Distribution Calculator

How to use this SMp(x) distribution calculator

What this calculator does

This calculator evaluates a six-parameter, two-branch SMp(x) function at a chosen value of x. It also integrates the full piecewise curve to show the area under the model and checks the special symmetric normalized conditions used here for cumulative probability calculations. The tool is useful for exploring how lower and upper bounds, a peak location, two shape powers, and a peak height affect a bounded probability-shaped function. It does not fit parameters from observed data, estimate uncertainty, or prove that an SMp(x) specification is the best statistical model for a dataset.

When to use it

Use it when you need to evaluate a proposed SMp(x) specification at a particular x, when you want to verify that a symmetric parameter set integrates to one, when you are studying how p₁ and p₂ change the rising and falling sides, or when you want a cumulative probability for a valid symmetric normalized setup. For general background, the NIST explanation of probability density and cumulative distribution functions describes the relationship between a density and the CDF obtained by integrating it.

How to calculate

1. The calculator opens with a complete demonstration model: PXmin = 0, Xmax = 10, ML = 5, p₁ = 2, p₂ = 2, Max = 0.3, and x = 3. Its Excel workbook is validated at startup, so Download Excel is immediately available.
2. Replace the six model parameters with your values. Use plain decimal notation with a period as the decimal separator. Scientific notation, thousands separators, percent signs, and other unit symbols are rejected rather than silently reinterpreted.
3. Enter Independent variable x. The function is zero outside the support, follows the rising power branch from PXmin through ML, and follows the falling branch from ML through Xmax.
4. Choose I want to calculate... and enter x₁. If you choose the between event, also enter x₂. Event probability is shown only when the calculator's probability-ready conditions are all satisfied.
5. Read SMp(x), the area, active branch, status, and event probability. Use Download Excel for a binary .xlsx workbook containing current inputs and outputs. Reset clears the demonstration and all semantic input values; the export is then disabled until you enter a complete valid model again.

Input guide

Lower limit value of x (PXmin) is a required finite decimal marking the left support boundary; it must be below ML. The startup example is 0. Increasing PXmin while keeping the other parameters fixed narrows the left side and changes its integrated area. A common mistake is setting PXmin equal to or above ML. Upper limit value of x (Xmax) is the required right boundary and must exceed ML; the example is 10. Moving Xmax changes the right-side width and total area.

x where SMp(x) = Max (ML) is the required peak location and must sit strictly between the limits; the example is 5. In the symmetric probability-ready case it equals the midpoint of PXmin and Xmax. Power (p₁) and Power (p₂) are required positive decimals controlling curvature on the rising and falling sides; both are 2 in the demonstration. Larger powers generally keep values lower away from the peak and make the rise or fall more concentrated near ML. Zero and negative powers are invalid.

Maximum of the model (Max) is the required positive peak height; the demonstration uses 0.3. A density value is not inherently limited to 1, so this field is not capped at 1. For the symmetric normalized case used for cumulative probabilities, Max must equal (p₁ + 1) / [2(ML – PXmin)]. Independent variable x is the required evaluation point, with 3 in the example; values outside the support are allowed and correctly return zero.

I want to calculate... is a required select control with three event definitions: P(X ≤ x₁), P(X ≥ x₁), or P(x₁ ≤ X ≤ x₂). x₁ is a required finite decimal event boundary; the example is 3. x₂ is required only for the between event, uses the same x units, and must be at least x₁; its prefilled demonstration value is 7. Do not reverse the two between-event bounds.

Output guide

SMp(x) is the exact piecewise function value at the current x, formatted as a decimal. Event probability is the integrated probability for the selected event and is formatted as a percentage; it is an integral-based estimate from this model, not an observed frequency. Area under model is the full integral from PXmin to Xmax; a value of 1 indicates normalization. Support width is Xmax – PXmin. Peak location repeats the canonical ML value so the model center is visible at a glance. Probability-ready status reports Yes/Ready only when PXmin ≥ 0, p₁ = p₂ > 1, ML is the support midpoint, and the area equals 1 within numerical tolerance. Active branch identifies whether x is below support, on the rising branch, on the falling branch, or above support.

Worked example

With the startup values PXmin = 0, Xmax = 10, ML = 5, p₁ = p₂ = 2, Max = 0.3, and x = 3, the rising formula applies. The ratio (3 – 0) / (5 – 0) is 0.6; squaring it gives 0.36; multiplying by 0.3 gives SMp(3) = 0.108. Each side has area 0.5 because 0.3 × 5 / (2 + 1) = 0.5, so the full area is 1. For P(X ≤ 3), integrating the rising branch from 0 to 3 gives 0.108, displayed as 10.80%. This is the same example stored in the initial workbook checkpoints.