Irwin-Hall sum of n uniform variables (PDF and CDF and moments)
UniformSumDistribution[n]
UniformSumDistribution[n, {min, max}]
PDF[UniformSumDistribution[2], x]
→ Piecewise[{{0, x < 0}, {x, Inequality[0, LessEqual, x, Less, 1]}, {2 - x, 1 <= x <= 2}}, 0]CDF[UniformSumDistribution[2], x]
→ Piecewise[{{0, x < 0}, {x^2/2, Inequality[0, LessEqual, x, Less, 1]}, {1 - (2 - x)^2/2, 1 <= x <= 2}}, 1]{Mean[UniformSumDistribution[n]], Variance[UniformSumDistribution[n]]}
→ {n/2, n/12}