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The article Three Basic Distributions from AnyDice shows the distributions (and cumulative distributions) of 1d10, 2d10 and 3d10.

Looking at the graphs, it seems there is a pattern here:

  • 1d10: the distribution is a piecewise polynomial with 1 piece of degree 0 (a constant)
  • 2d10: the distribution is a piecewise polynomial with 2 pieces of degree 1 (2 linear functions)
  • 3d10: the distribution may be a piecewise polynomial with 3 pieces of degree 2

Therefore it seems like distributions of dice pools are piecewise polynomials.
Is it true?

Is there a general formula for the distribution of dice pools?

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In the continuous case this is the Irwin-Hall distribution which indeed has a pdf which can be written as a spline.

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