最近在做仿真模拟的过程中，发现包裹正态分布并不适用于某些场景，在搜索过程中就发现了本博文中所要描述的这种分布，即 Irwin–Hall distribution（要求所涉及的均匀分布之间是相互独立的）
值得注意的是，这与 Bates distribution ，即多个独立的均匀分布的均值的分布是不同的，详见https://github.com/d3/d3/issues/1647
假设 U k U_k Uk 服从标准的均匀分布，即 U k ∼ U ( 0 , 1 ) U_k\sim U(0,1) Uk∼U(0,1)，那么本分布的随机变量可表示为 X = ∑ k = 1 n U k X=\sum_{k=1}^{n} U_{k} X=k=1∑nUk 它的概率密度函数可以表示为 f X ( x ; n ) = 1 2 ( n − 1 ) ! ∑ k = 0 n ( − 1 ) k ( n k ) ( x − k ) n − 1 sgn ( x − k ) f_{X}(x ; n)=\frac{1}{2(n-1) !} \sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l} n \\ k \end{array}\right)(x-k)^{n-1} \operatorname{sgn}(x-k) fX(x;n)=2(n−1)!1k=0∑n(−1)k(nk)(x−k)n−1sgn(x−k) 其中 x ∈ [ 0 , n ] x\in[0, n] x∈[0,n] sgn ( x − k ) = { − 1 x < k 0 x = k 1 x > k \operatorname{sgn}(x-k)=\left\{\begin{array}{ll} -1 & x<k \\ 0 & x=k \\ 1 & x>k \end{array}\right. sgn(x−k)=⎩⎨⎧−101x<kx=kx>k
它的均值和方差分别为 n 2 \dfrac{n}{2} 2n和 n 12 \dfrac{n}{12} 12n
下面我们再列举一些常用表达
n = 2 n=2 n=2 f X ( x ) = { x 0 ≤ x ≤ 1 2 − x 1 ≤ x ≤ 2 f_{X}(x)=\left\{\begin{array}{ll} x & 0 \leq x \leq 1 \\ 2-x & 1 \leq x \leq 2 \end{array}\right. fX(x)={x2−x0≤x≤11≤x≤2 n = 3 n=3 n=3 f X ( x ) = { 1 2 x 2 0 ≤ x ≤ 1 1 2 ( − 2 x 2 + 6 x − 3 ) 1 ≤ x ≤ 2 1 2 ( x − 3 ) 2 2 ≤ x ≤ 3 f_{X}(x)=\left\{\begin{array}{ll} \frac{1}{2} x^{2} & 0 \leq x \leq 1 \\ \frac{1}{2}\left(-2 x^{2}+6 x-3\right) & 1 \leq x \leq 2 \\ \frac{1}{2}(x-3)^{2} & 2 \leq x \leq 3 \end{array}\right. fX(x)=⎩⎨⎧21x221(−2x2+6x−3)21(x−3)20≤x≤11≤x≤22≤x≤3 n = 4 n=4 n=4 f X ( x ) = { 1 6 x 3 0 ≤ x ≤ 1 1 6 ( − 3 x 3 + 12 x 2 − 12 x + 4 ) 1 ≤ x ≤ 2 1 6 ( 3 x 3 − 24 x 2 + 60 x − 44 ) 2 ≤ x ≤ 3 1 6 ( − x + 4 ) 3 3 ≤ x ≤ 4 f_{X}(x)=\left\{\begin{array}{ll} \frac{1}{6} x^{3} & 0 \leq x \leq 1 \\ \frac{1}{6}\left(-3 x^{3}+12 x^{2}-12 x+4\right) & 1 \leq x \leq 2 \\ \frac{1}{6}\left(3 x^{3}-24 x^{2}+60 x-44\right) & 2 \leq x \leq 3 \\ \frac{1}{6}(-x+4)^{3} & 3 \leq x \leq 4 \end{array}\right. fX(x)=⎩⎪⎪⎨⎪⎪⎧61x361(−3x3+12x2−12x+4)61(3x3−24x2+60x−44)61(−x+4)30≤x≤11≤x≤22≤x≤33≤x≤4