How this calculator works
The left branch rises as a power of distance from the lower limit and the right branch falls as a power of distance from the upper limit. This implementation uses the lower zero of the curve as the support boundary. It integrates continuous branches or sums integer weights to choose a normalising height. Supplying Max changes the raw curve and total area or mass; all reported probabilities and moments still use the normalised curve. This specialised family has limited literature and is not a clinical prediction tool.
Formula
w(x) = ((x-a)/(m-a))^p1 for a≤x≤m; w(x) = ((b-x)/(b-m))^p2 for m<x≤b; zero outside. Continuous A = (m-a)/(p1+1)+(b-m)/(p2+1); normalised f = w/A. Discrete A = Σw(k).
Worked example
For a = 0, b = 2, m = 1 and both powers 1, the continuous model is triangular: A = 1, peak density = 1, mean = 1, variance = 1/6 and F(1) = 0.5.
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Frequently asked questions
Is an arbitrary Max a probability density?
Only if its total area is one. The calculator reports raw area and separately normalises the probabilities.
What changes in discrete mode?
Normalisation is a sum over integer values, rather than a continuous integral. The same branch shape can therefore have different moments.
Is this a replacement for standard distributions?
No general superiority is assumed. Use an applied model only with evidence supporting its shape and assumptions.