The SMp(x) distribution calculator above evaluates the density and the cumulative probability of the SMp family at any point, computes the normalising constant that makes the total area equal one, and returns the mean, variance and standard deviation. It also reports the total area for any Max you supply, so you can see immediately whether a chosen peak height gives a genuine probability density or not.
Arb Digital publishes it with a caveat stated in the first paragraph rather than buried. SMp is not a standard distribution in the sense that the normal, Weibull or Rayleigh laws are. It is a specific parametric family introduced in a single line of work, with a small literature, and it is not implemented in the mainstream statistical libraries. This page computes it faithfully and cites the source, but it does not present it as an established alternative to the distributions in our normal distribution calculator or beta distribution calculator.
What This SMp Distribution Calculator Does
The family is defined on a bounded interval from PXmin to Xmax, with a peak at a point ML in between. Two powers, p₁ and p₂, control the curvature of the left and right branches independently, and Max is the height of the peak. With six parameters and a bounded support it is capable of a wide range of unimodal shapes, including strongly asymmetric ones.
The calculator takes all six, or five of them plus an automatic Max, and evaluates the function. Because a probability density must integrate to one, only one value of Max is admissible for any given choice of the other five, and the default setting computes it. If you supply your own Max instead, the tool computes and displays the resulting area so you can see the discrepancy rather than being told nothing.
Alongside the density and cumulative probability it returns the mean, the variance, the standard deviation and the mode. All of these have exact closed forms for this family, derived by integrating the two power-law branches, so nothing here relies on numerical integration.
How to Use It
- Set the support with PXmin and Xmax, and place the peak at ML strictly between them. All three are in the units of your variable.
- Choose the two powers. Values above 1 make the branch concave towards the peak; values below 1 make it convex. Setting both to 1 gives straight lines and the ordinary triangular distribution.
- Leave Max on automatic unless you specifically want to test a supplied value. On automatic the result is a proper density; on manual it may not be.
- Enter a value of x. Outside the support the density is zero and the tool says so explicitly.
- Check the total area figure. It should read exactly 1 on the automatic setting, and it is displayed as a running verification rather than assumed.
The Formula and How It Is Calculated
The function is a two-branch power law, zero outside the interval. On the left branch, for x between PXmin and ML, it is Max multiplied by ((x − PXmin) / (ML − PXmin)) raised to the power p₁. On the right branch, for x between ML and Xmax, it is Max multiplied by ((Xmax − x) / (Xmax − ML)) raised to the power p₂. Both branches equal Max at x = ML and zero at the respective endpoint.
Integrating each branch is elementary. The left contributes Max × (ML − PXmin) / (p₁ + 1) and the right contributes Max × (Xmax − ML) / (p₂ + 1). Setting the total to one gives Max as the reciprocal of the sum of those two bracketed terms without the Max, which is what the automatic setting computes. In the symmetric case, where ML sits at the midpoint and both powers are equal, that reduces to Max = (p₁ + 1) / (2 (ML − PXmin)), which is the normalisation condition stated in the defining paper and is a useful cross-check on any implementation.
The cumulative function follows by integrating from the lower limit. Below ML it is Max × (ML − PXmin) / (p₁ + 1) multiplied by the left-branch ratio raised to the power p₁ + 1. Above ML it is that value at ML, plus the right branch's total contribution multiplied by one minus the right-branch ratio raised to the power p₂ + 1. At x = Xmax it evaluates to exactly one.
Worked example with the defaults: PXmin 0, Xmax 10, ML 4, p₁ 2, p₂ 3, evaluated at x = 3. The left integral term is 4 divided by 3, which is 1.3333333, and the right is 6 divided by 4, which is 1.5, so Max is the reciprocal of 2.8333333, which is 0.35294118. At x = 3 the left-branch ratio is 0.75, and 0.75 squared is 0.5625, so the density is 0.35294118 times 0.5625, which is 0.19852941. The cumulative probability is 0.35294118 times 1.3333333 times 0.75 cubed, which is 0.47058824 times 0.421875, or 0.19852941 again. The mean works out to 4.16470588, slightly above the peak at 4 because the longer right tail pulls it that way.
Where This Definition Comes From
The family was introduced by Terman Frometa-Castillo in the chapter "The SMp(x or y;PXmin,Xmax,ML,p1,p2,Max) a Probabilistic Distribution, or a Probability Density Function of a Random Variable X", published by Springer in the IFMBE Proceedings series, with the digital object identifier 10.1007/978-3-030-14070-0_48. The stated motivation is modelling in radiotherapy, where bounded quantities with a most-likely value and asymmetric behaviour on either side of it occur naturally.
The construction takes the triangular distribution as its starting point and generalises the two straight-line branches to power laws. That lineage is visible in the arithmetic: set both powers to 1 and every formula on this page reduces exactly to the standard triangular results, including the peak height of 2 divided by the range for a symmetric case, which you can check against the MathWorld entry on the triangular distribution. That reduction is the strongest available verification that an implementation of SMp is correct, and it is why the triangular case is offered as a button here.
It is worth being clear about the status of this family. It has a small literature relative to the standard distributions, it is associated primarily with one line of work, and it is not implemented in the major statistical software packages. Anyone considering it for applied work should read the source and satisfy themselves that it fits their problem better than the well-established alternatives, rather than adopting it because a calculator exists for it.
How the Powers Change the Shape
Each power controls one branch and nothing else, which is the main practical attraction of the family. A power of exactly 1 gives a straight line from the endpoint up to the peak. A power greater than 1 gives a curve that hugs the axis near the endpoint and rises steeply near the peak, concentrating mass around ML. A power less than 1 does the opposite: the curve rises sharply from the endpoint and flattens as it approaches the peak, spreading mass towards the tail.
Because the two powers are independent, the family can produce shapes that a symmetric two-parameter distribution cannot: a steep rise on one side and a long, slowly decaying approach on the other. That flexibility comes at the cost of parameters. Six of them, with two determined by the support and one determined by normalisation, still leaves three free shape parameters for what is a unimodal bounded density, which is more than the beta distribution needs for a comparable range of shapes.
One structural limitation is worth noting. Because both branches are pure powers of the distance from the endpoint, the density is always exactly zero at both PXmin and Xmax when the powers exceed zero, and always has its single maximum exactly at ML. The family cannot produce a bimodal shape, a U shape, or a density with non-zero mass at an endpoint. The beta distribution calculator covers the bounded cases that include U shapes, and the uniform distribution calculator the flat case.
Reading Density, Area and the Max Parameter
Max is a height, not a probability. On a narrow support it will be large: squeeze the same unit of area into a range of 0.1 and the peak must exceed 10. Seeing a Max above one is not an error and does not mean anything is wrong.
The total area figure is the check that matters. It is computed independently from the two integral terms and displayed on every run. On the automatic setting it reads exactly 1, up to floating-point rounding. On the manual setting it reads whatever your Max produces, and if that is not 1 then the function you have specified is not a probability density: its cumulative values will not reach one, and the mean and variance reported will be those of the normalised version rather than of what you entered.
The mean, variance and standard deviation are always computed from the normalised density, because those quantities are only defined for a proper distribution. The tool states this in the working panel whenever the supplied Max does not normalise. For the general question of reading densities against probabilities, our Rayleigh distribution calculator covers the same distinction on an unbounded support.
Arb Digital builds measurement and reporting systems where every model is documented and every figure is traceable to where it came from.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Choosing Max freely — only one value normalises the function for a given support and pair of powers. Any other value makes it something other than a probability density.
- Placing ML at an endpoint — the family requires ML strictly inside the interval, because one branch would otherwise have zero width and the ratio defining it would be a division by zero.
- Using a power of zero — that makes a branch constant rather than rising to a peak, which is outside the family as defined and produces a discontinuous function.
- Treating SMp as an established standard — it has a small literature and no mainstream software implementation, so any applied use needs the source read and the fit justified against ordinary alternatives.
- Reading the density as a probability — it is probability per unit of x and routinely exceeds one on a narrow support. Only the area between two points is a probability.
Related Free Tools From Arb Digital
Cover the standard bounded family with the beta distribution calculator, the flat case with the uniform distribution calculator, the unbounded right-skewed case with the Weibull distribution calculator or the Rayleigh distribution calculator, and the symmetric unbounded case with the normal distribution calculator. The full free online tools hub lists every statistics tool we publish.
Frequently Asked Questions
A bounded, unimodal family defined by two power-law branches meeting at a peak. It generalises the triangular distribution by replacing each straight branch with a power of the distance from the endpoint.
In a Springer chapter by Terman Frometa-Castillo with the identifier 10.1007/978-3-030-14070-0_48, published in the IFMBE Proceedings series and motivated by modelling problems in radiotherapy.
PXmin and Xmax bound the support, ML is where the density peaks, p₁ and p₂ are the powers of the left and right branches, and Max is the height of the peak.
Not if you want a probability density. Exactly one value makes the area under the curve equal one, and this page computes it by default while still reporting the area for any value you supply.
Set both powers to 1. The family reduces exactly to the triangular distribution, whose density, cumulative function and mean can be checked against any standard reference.
No. It has a small literature, is associated with one line of work, and is not implemented in the major statistical packages. Applied use should be justified against established alternatives.
No. Both branches are monotonic powers of the distance from an endpoint, so the density has exactly one maximum, always at ML, and is zero at both ends of the support.
Both are bounded and flexible, but beta achieves a comparable range of unimodal shapes with two shape parameters and can also produce U shapes, which SMp cannot.
This page implements the SMp(x) density as defined in the cited source and is provided for educational purposes. The family is not a mainstream standard, and choosing it for applied work should follow from reading that source and testing the fit against established distributions rather than from the availability of this calculator.
