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随机数

随机数是指从某个特定分布中随机选择的数字，选择大量这样的数字可以重现潜在的分布。几乎总是，这些数字还被要求是独立的，以便连续的数字之间没有相关性。计算机生成的随机数有时被称为伪随机数，而术语“随机”则保留给不可预测的物理过程的输出。当不加限定地使用时，“随机”一词通常意味着“具有均匀分布的随机”。当然，其他分布也是可能的。例如，Box-Muller 变换允许将成对的均匀随机数转换为具有二维正态分布的相应随机数。

不可能生成任意长的随机数字字符串并证明它是随机的。奇怪的是，人类也很难生成随机数字字符串，并且可以编写计算机程序，这些程序平均而言实际上可以根据之前的数字预测人类将写下的一些数字。

有许多常用的方法用于生成伪随机数，其中最简单的方法是线性同余法。另一种简单而优雅的方法是元胞自动机规则 30，其中心列由 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, ... (OEIS A051023) 给出，它提供了 Wolfram 语言中用于大整数的随机数生成器。大多数随机数生成器需要指定一个用作起点的初始数字，这被称为“种子”。可以通过检查其噪声球来分析给定算法生成的随机数的优劣。

当在某个指定的边界上生成随机数时，通常需要对分布进行归一化，以便每个微分面积都被均匀填充。例如，从均匀分布中选取 theta 和 phi 不能给出球体点拾取的均匀分布。

为了从均匀分布 P(y) 生成幂律分布 P(x)，对于 x in [x_0,x_1]，写成 P(x)=Cx^n。然后归一化给出

int_(x_0)^(x_1)P(x)dx=C([x^(n+1)]_(x_0)^(x_1))/(n+1)=1,
(1)

所以

C=(n+1)/(x_1^(n+1)-x_0^(n+1)).
(2)

设 Y 是 [0,1] 上的均匀分布变量。那么

D(x)=int_(x_0)^xP(x^')dx^'
(3)
=Cint_(x_0)^xx^('n)dx^'
(4)
=C/(n+1)(x^(n+1)-x_0^(n+1))
(5)
=y,
(6)

且变量由下式给出

X=((n+1)/Cy+x_0^(n+1))^(1/(n+1))
(7)
=[(x_1^(n+1)-x_0^(n+1))y+x_0^(n+1)]^(1/(n+1))
(8)

分布为 P(x)。

另请参阅

Bays' Shuffle, Box-Muller 变换, Cliff 随机数生成器, 准随机序列, 随机变量, Schrage 算法, 随机, 均匀分布

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参考文献

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随机数

请引用为

Weisstein, Eric W. "随机数。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/RandomNumber.html

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