朴素解不起作用的原因是它给了靠近圆中心的点更高的概率密度。换句话说，半径为r/2的圆被选中点的概率为r/2，但它的面积(点的数量)为*r^2/4。

因此，我们希望半径概率密度具有以下性质:

选择半径小于或等于给定r的概率必须与半径为r的圆的面积成正比(因为我们希望在点上有一个均匀的分布，面积越大意味着点越多)。

换句话说，我们希望在[0,r]之间选择半径的概率等于它在圆的总面积中所占的份额。圆的总面积是*R^2，半径为R的圆的面积是*R^2。因此，我们希望在[0,r]之间选择半径的概率为(pi*r^2)/(pi* r^2) = r^2/ r^2。

现在来算算:

The probability of choosing a radius between [0,r] is the integral of p(r) dr from 0 to r (that's just because we add all the probabilities of the smaller radii). Thus we want integral(p(r)dr) = r^2/R^2. We can clearly see that R^2 is a constant, so all we need to do is figure out which p(r), when integrated would give us something like r^2. The answer is clearly r * constant. integral(r * constant dr) = r^2/2 * constant. This has to be equal to r^2/R^2, therefore constant = 2/R^2. Thus you have the probability distribution p(r) = r * 2/R^2

Note: Another more intuitive way to think about the problem is to imagine that you are trying to give each circle of radius r a probability density equal to the proportion of the number of points it has on its circumference. Thus a circle which has radius r will have 2 * pi * r "points" on its circumference. The total number of points is pi * R^2. Thus you should give the circle r a probability equal to (2 * pi * r) / (pi * R^2) = 2 * r/R^2. This is much easier to understand and more intuitive, but it's not quite as mathematically sound.

朴素解不起作用的原因是它给了靠近圆中心的点更高的概率密度。换句话说，半径为r/2的圆被选中点的概率为r/2，但它的面积(点的数量)为*r^2/4。

因此，我们希望半径概率密度具有以下性质:

选择半径小于或等于给定r的概率必须与半径为r的圆的面积成正比(因为我们希望在点上有一个均匀的分布，面积越大意味着点越多)。

换句话说，我们希望在[0,r]之间选择半径的概率等于它在圆的总面积中所占的份额。圆的总面积是*R^2，半径为R的圆的面积是*R^2。因此，我们希望在[0,r]之间选择半径的概率为(pi*r^2)/(pi* r^2) = r^2/ r^2。

现在来算算:

The probability of choosing a radius between [0,r] is the integral of p(r) dr from 0 to r (that's just because we add all the probabilities of the smaller radii). Thus we want integral(p(r)dr) = r^2/R^2. We can clearly see that R^2 is a constant, so all we need to do is figure out which p(r), when integrated would give us something like r^2. The answer is clearly r * constant. integral(r * constant dr) = r^2/2 * constant. This has to be equal to r^2/R^2, therefore constant = 2/R^2. Thus you have the probability distribution p(r) = r * 2/R^2

Note: Another more intuitive way to think about the problem is to imagine that you are trying to give each circle of radius r a probability density equal to the proportion of the number of points it has on its circumference. Thus a circle which has radius r will have 2 * pi * r "points" on its circumference. The total number of points is pi * R^2. Thus you should give the circle r a probability equal to (2 * pi * r) / (pi * R^2) = 2 * r/R^2. This is much easier to understand and more intuitive, but it's not quite as mathematically sound.

半径和“靠近”该半径的点的数量之间存在线性关系，因此他需要使用半径分布，这也使得半径r附近的数据点的数量与r成正比。

下面是我的Python代码，从半径为rad的圆中生成num个随机点:

import matplotlib.pyplot as plt
import numpy as np
rad = 10
num = 1000

t = np.random.uniform(0.0, 2.0*np.pi, num)
r = rad * np.sqrt(np.random.uniform(0.0, 1.0, num))
x = r * np.cos(t)
y = r * np.sin(t)

plt.plot(x, y, "ro", ms=1)
plt.axis([-15, 15, -15, 15])
plt.show()

如何在半径为R的圆内随机生成一个点:

r = R * sqrt(random())
theta = random() * 2 * PI

(假设random()均匀地给出0到1之间的值)

如果你想把它转换成笛卡尔坐标，你可以做到

x = centerX + r * cos(theta)
y = centerY + r * sin(theta)

为什么sqrt(随机())?

让我们看看sqrt(random())之前的数学运算。为简单起见，假设我们是在单位圆上工作，即R = 1。

点与点之间的平均距离应该是相同的，不管我们看的距离中心有多远。这意味着，例如，观察一个周长为2的圆的周长，我们应该找到的点的数量是周长为1的圆周长上点的数量的两倍。

由于圆的周长(2πr)随r线性增长，因此随机点的数量应该随r线性增长。换句话说，期望的概率密度函数(PDF)线性增长。由于PDF的面积应该等于1，最大半径是1，我们有

所以我们知道随机值的理想密度应该是什么样的。 现在:当我们只有一个0到1之间的均匀随机值时，我们如何生成这样一个随机值?

我们用了一个叫做反变换采样的技巧

从PDF中创建累积分布函数(CDF) 沿着y = x镜像 将得到的函数应用于0到1之间的统一值。

听起来复杂吗?让我插入一段带有小侧轨的引语来传达直觉:

Suppose we want to generate a random point with the following distribution:                  That is 1/5 of the points uniformly between 1 and 2, and 4/5 of the points uniformly between 2 and 3. The CDF is, as the name suggests, the cumulative version of the PDF. Intuitively: While PDF(x) describes the number of random values at x, CDF(x) describes the number of random values less than x. In this case the CDF would look like:                  To see how this is useful, imagine that we shoot bullets from left to right at uniformly distributed heights. As the bullets hit the line, they drop down to the ground:                  See how the density of the bullets on the ground correspond to our desired distribution! We're almost there! The problem is that for this function, the y axis is the output and the x axis is the input. We can only "shoot bullets from the ground straight up"! We need the inverse function! This is why we mirror the whole thing; x becomes y and y becomes x:                  We call this CDF-1. To get values according to the desired distribution, we use CDF-1(random()).

所以，回到生成随机半径值，其中PDF等于2x。

步骤1:创建CDF: 由于我们处理的是实数，CDF表示为PDF的积分。

CDF（x） = ∫ 2x = x2

步骤2:沿y = x镜像CDF:

从数学上讲，这可以归结为交换x和y并求解y:

CDF: y = x2 交换:x = y2 解:y =√x CDF-1: y =√x

步骤3:将得到的函数应用于0到1之间的统一值

CDF-1(random()) =√random()

这就是我们要推导的:-)

你也可以用你的直觉。

圆的面积是*r^2

为r = 1

得到的面积是。让我们假设我们有某种函数f，它将N=10个点均匀分布在一个圆内。这个比值是10 /

现在我们把面积和点数翻倍

对于r=2和N=20

面积是4，比值是20/4或10/2。半径越大，比值就越小，因为它的增长是二次的，N是线性的。

为了解决这个问题，我们可以说

x = r^2
sqrt(x) = r

如果在极坐标下生成一个向量

length = random_0_1();
angle = random_0_2pi();

更多的点会落在中心周围。

length = sqrt(random_0_1());
angle = random_0_2pi();

长度不再是均匀分布的，但是向量现在是均匀分布的。