理解“随机性" [英] Understanding "randomness"

问题描述

我无法理解这个，哪个更随机?

rand()

或:

rand() * rand()

我发现这是一个真正的脑筋急转弯，你能帮我吗?

<小时>

直觉上我知道数学答案将是它们同样随机，但我不禁想到，如果将两者相乘时运行随机数算法"两次，您将创建更随机的东西而不是只做一次.

解决方案

澄清一下

尽管当您尝试发现伪随机变量或其乘法的随机性时，前面的答案是正确的，但您应该注意，虽然 Random() 通常是均匀分布的，Random() * Random() 不是.

示例

这是一个通过伪随机变量模拟的指出Random()随着术语的增加趋于 在评论中指出最后两张图片中显示的概率分布是称为 Irwin-Hall 分布

感谢 Heike 的精彩torn[] 函数

I can't get my head around this, which is more random?

rand()

OR:

rand() * rand()

I´m finding it a real brain teaser, could you help me out?

---

EDIT:

Intuitively I know that the mathematical answer will be that they are equally random, but I can't help but think that if you "run the random number algorithm" twice when you multiply the two together you'll create something more random than just doing it once.

解决方案

Just a clarification

Although the previous answers are right whenever you try to spot the randomness of a pseudo-random variable or its multiplication, you should be aware that while Random() is usually uniformly distributed, Random() * Random() is not.

Example

This is a uniform random distribution sample simulated through a pseudo-random variable:

        BarChart[BinCounts[RandomReal[{0, 1}, 50000], 0.01]]

While this is the distribution you get after multiplying two random variables:

        BarChart[BinCounts[Table[RandomReal[{0, 1}, 50000] * 
                                 RandomReal[{0, 1}, 50000], {50000}], 0.01]]

So, both are "random", but their distribution is very different.

Another example

While 2 * Random() is uniformly distributed:

        BarChart[BinCounts[2 * RandomReal[{0, 1}, 50000], 0.01]]

Random() + Random() is not!

        BarChart[BinCounts[Table[RandomReal[{0, 1}, 50000] + 
                                 RandomReal[{0, 1}, 50000], {50000}], 0.01]]

The Central Limit Theorem

The Central Limit Theorem states that the sum of Random() tends to a normal distribution as terms increase.

With just four terms you get:

BarChart[BinCounts[Table[RandomReal[{0, 1}, 50000] + RandomReal[{0, 1}, 50000] +
                   Table[RandomReal[{0, 1}, 50000] + RandomReal[{0, 1}, 50000],
                   {50000}],
         0.01]]  

And here you can see the road from a uniform to a normal distribution by adding up 1, 2, 4, 6, 10 and 20 uniformly distributed random variables:

Edit

A few credits

Thanks to Thomas Ahle for pointing out in the comments that the probability distributions shown in the last two images are known as the Irwin-Hall distribution

Thanks to Heike for her wonderful torn[] function

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