Python随机生成器不是那么随机吗？

Question

Is the python random generator considered "good"? As in, does it simulate randomness very well? I made a small program which simulates a person starting at (0, 0) and taking a random step either east, west, north and south. As you run simulations with larger and larger number of steps, you would expect the final position of the person to be closer and closer to the origin. However, as I increased the number of steps in my simulation, the final position was getting farther and farther away from the origin. The program I used is here:

import random

def walk():
    north = 0
    east = 0
    for i in range(10000):
        direction = random.randint(0, 100)
    if direction <= 25:
        north = north + 1
    elif direction > 25 and direction <= 50:
        north = north - 1
    elif direction > 50 and direction <= 75:
        east = east + 1
    else:
        east = east - 1
return north, east



def main():
    for i in range(500):
        north, east = walk()
        print("NE : ", north, east)


main()        

Answer 1

As you run simulations with larger and larger number of steps, you would expect the final position of the person to be closer and closer to the origin.

This simply is not true. Imagine you start at (0,0). You take one step east, taking you to (1,0). Now, in order to get closer to the origin, you need to take a step west. Taking a step in any other direction will take you further from the origin. So, you have a 1/4 chance of going closer to the origin and a 3/4 chance of going further away from the origin. This is true no matter where you are; more often than not, a step will take you further from the origin. Thus, your simulation is behaving as expected; the longer you go, the further from the origin you get.

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To see why this is true; consider a one-dimensional walk, where you can step left or right. If you take ten steps, the expected value is 50:50 left and right; 5 left and 5 right, taking you back to the origin. Of course, you might end up with 60:40 or 70:30, leaving you one or two steps away from the origin. It is a small sample size, after all. If you take 1000 steps, you might end up with something like 520 left to 480 right, pretty close to that 50:50 ratio. If you take 10000 steps, you'll be even closer to the 50:50 ratio, perhaps with something like 5050:4950.

However, there is something very important to note; as your sample size gets larger, the proportion of left to right steps is closer to 50:50, but the absolute difference in number between left and right steps gets larger . In that last case, you have a 50.5:49.5 ratio, but you're fifty steps away from the origin, compared to your one step away from the origin in the case where you have a 60:40 ratio with ten steps.

Answer 2

It's not python, it's this line:

        direction = random.randint(0, 100)

The above will generate a value 0 <= N <= 100, which is 101 possible values. You have introduced bias into your generator, and given the bias, with larger numbers you will in fact stray further and further from the origin.

With an unbiased generator, your mean value is zero.

Answer 3

Whatever the above answers, which are all good and correct, the random generator implemented in Python is the Mersenne Twister one which is known to not pass 2 of the hardest tests of the famous application TestU01 and which is known also to be a little big long to evaluate. You could rather try this library: PyRandLib. See:

• https://schmouk.github.io/PyRandLib/ to easily download archives versions, and
• https://github.com/schmouk/PyRandLib to get access to code.

This library contains many of the best-in-class pseudo-random numbers generators while acting exactly as does the Python "built-in" library random. Just un-zip or un-tar the downloaded archive in the 'Lib/site-packages/' sub-directory of your Python directory. And then, enjoy :-)