Let’s do some real calculation and uncover the myths of statistics…

From random walk to binomial distribution

1-D Random Walk $…-2\leftrightharpoons-1\leftrightharpoons0\leftrightharpoons1\leftrightharpoons2…$ with rightward transition prob as $p(n)=p$ and leftward prob as $q(n)=1-p=q$​. After $N$ steps, $P(n_N \vert n_0=0) = \binom{N}{\frac{N+n}{2}} p^{\frac{N+n}{2}} (1-p)^{\frac{N-n}{2}}$ obeys a binomial distribution. (Compare with n Bernoulli trials, here if you lose, you get $-1$ instead of $0$).

Gaussian distribution as approximation of binomial distribution

From binomial distribution $X\sim B(n,p)$, $\Pr(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$, using Stirling approximation and Taylor expansion, we get $X\sim \mathcal N(np, np(1-p))$, the Gaussian distribution.

• 对时间做一阶展开，然后用傅立叶变换法求解扩散方程。

$P(x, t) = \frac{1}{\sqrt{4\pi Dt}}\exp[-\frac{(x-vt)^2}{4Dt}]$，其中 $v = \frac{a}{\tau}(2p-1)$, $D = \frac{a^2}{2\tau}$; $t = N\tau$, $x = na$。