# Random walk in the plane

A particles moves in $\mathbb{R^2}$ started at the origin. At each stage $i (i = 1, 2, ...)$, the particles would move, independently of all the stages before, one of the four directions North, East, South and West 1 unit, with probability $1 \over 4$ each.Let $T_n$ be the distance from the origin just after $n$ steps. What is $E(T_n)$. I tried define 4 $1\times 2$matrice with only $1,0,-1$ in all entries. The use technique similar to simple random walk in 1 dimension and we know that for a $1\times 2$ matrice $(x,y)$, the distance from origin =$\sqrt {x^2+y^2}$ but i am not sure how to like the 1 D to 2 D. As in 2 D there are 4 possible direction.

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According to this, it takes $({d\over l})^2$ steps on average to travel a distance $d$ with step length $l$. –  Ben Nov 1 '12 at 11:50

$$\sqrt{2n} \dfrac{\Gamma(\frac{3}{2})}{\Gamma(\frac{1}{2})}=\sqrt{\frac{n}{2}}$$
Careful with that axe, Eugene: This is not an exact formula but an asymptotics when $n\to\infty$. Except in 1D, there is no known formula, valid for every $n$, other than the cumbersome summations one can fathom. –  Did Nov 2 '12 at 7:56
I guess this formula holds only if we consider a very large $n$ where $n\rightarrow\infty$ but what about for a small $n$ –  Mathematics Nov 2 '12 at 8:13