正态分布随机数

时间关系暂时先不介绍啥是正态分布了，网上关于生成正态分布随机数的方法也有很多。

下面是moro 逆正态累积分布函数（Moro's Inverse Cumulative Normal Distribution function）

 

double MoroInvCND(double prob)
{
    const double a1 = 2.50662823884;
    const double a2 = -18.61500062529;
    const double a3 = 41.39119773534;
    const double a4 = -25.44106049637;
    const double b1 = -8.4735109309;
    const double b2 = 23.08336743743;
    const double b3 = -21.06224101826;
    const double b4 = 3.13082909833;
    const double c1 = 0.337475482272615;
    const double c2 = 0.976169019091719;
    const double c3 = 0.160797971491821;
    const double c4 = 2.76438810333863E-02;
    const double c5 = 3.8405729373609E-03;
    const double c6 = 3.951896511919E-04;
    const double c7 = 3.21767881768E-05;
    const double c8 = 2.888167364E-07;
    const double c9 = 3.960315187E-07;

    double z;

    bool negate = false;
    
	const double maxValue = static_cast(0xffffffffUL);

    // Ensure the conversion to floating point will give a value in the
    // range (0,0.5] by restricting the input to the bottom half of the
    // input domain. We will later reflect the result if the input was
    // originally in the top half of the input domain
	unsigned int x = (unsigned int)(prob * maxValue);
    if (x >= 0x80000000UL)
    {
        x = 0xffffffffUL - x;
        negate = true;
    }

    // x is now in the range [0,0x80000000) (i.e. [0,0x7fffffff])
    // Convert to floating point in (0,0.5]
    const double x1 = 1.0 / maxValue;
    const double x2 = x1 / 2.0;
    double p1 = x * x1 + x2;
    // Convert to floating point in (-0.5,0]
    double p2 = p1 - 0.5;

    // The input to the Moro inversion is p2 which is in the range
    // (-0.5,0]. This means that our output will be the negative side
    // of the bell curve (which we will reflect if "negate" is true).

    // Main body of the bell curve for |p| < 0.42
    if (p2 > -0.42)
    {
        z = p2 * p2;
        z = p2 * (((a4 * z + a3) * z + a2) * z + a1) / ((((b4 * z + b3) * z + b2) * z + b1) * z + 1.0);
    }
    // Special case (Chebychev) for tail
    else
    {
        z = log(-log(p1));
        z = - (c1 + z * (c2 + z * (c3 + z * (c4 + z * (c5 + z * (c6 + z * (c7 + z * (c8 + z * c9))))))));
    }

    // If the original input (x) was in the top half of the range, reflect
    // to get the positive side of the bell curve
    return negate ? -z : z;
}

那么如何使用呢：

MoroInvCND(0.5)

输出的是累积概率到0.5的那个对应的值，即0。

所以可以先随机生成（0，1）之间的随机数，然后用上面的函数将其转换成正态分布的随机数