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作业，存档一下过程

1、Task

​ Imagine a drunken man who, starting out leaning against a lamp post in the middle of an oPEn space, takes a series of steps of the same length: 1 meter . The direction of these steps is randoMLy chosen From North, South, East or West. After n steps, how far (*d*), generally speaking, is the man from the lamp post? Note that d is the Euclidean distance of the man from the lamp-post. Deduce the relationship.

2、Relationship Conclusion

​ Just a apPRoximation of the result.

3、Evidence (Mathematics deduction)

To get to the conclusion of

We consider the drunken man walking in a coordinate System and the lamp spot as the origin,

then we will get his posITion as (x,y)

and the distance will be

And we assume him walking on

​ West-East direction (x axis) for i steps

​ North-south direction (y axis) for k steps

Then we will have

If we see Xa/Ya represent the steps as -1/1 for opposite direction.

​ each XaXa pair will be within the following types:

​ and the probability of these pairs will be the same because it's Random

​ On average will be 0,

​ Therefore,

​ the same procedure may be easily adapted to Y²

So, we can approxiMATEly deduce that

QED

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