NIKOLA UBAVIĆ Srpski

# Random walk

Below is given one-dimensional random walk simulation. Particles start from $$0$$ and in every step each particle change its position by $$+1$$ with probability $$p$$ or by $$-1$$ with probability $$1-p$$.

 Number of experiments (particles) (1-10.000) Number of steps in each experiment (10-10.000) Probability $$p$$ (in percents) (0-100)
Start simulation

Particles trajectories:

Distribution of final positions:

Statistics:

 $$\mathrm E \left(S_n\right)$$ $$\sigma \left(S_n\right)$$ $$\min S_n$$ $$\max S_n$$ Experiment Model (see below)

## Mathematical model

Let $$Z_i$$ be random variable that equals to change of a particle position in step $$i$$. Variable $$Z_i$$ has probability distribution $\begin{pmatrix} -1 & 1 \\ 1-p & p\end{pmatrix}.$

If the particle starts from the $$0$$, then it's position after $$n$$ steps is given by the random variable $$S_n = Z_1 + Z_2 + \dots + Z_n$$. Therefore, expectation of the $$S_n$$ is given by $\mathrm E \left(S_n\right) = \sum_{i=1}^n \mathrm E \left(Z_i\right) = n\left(2p-1\right).$ Specially, if $$p=0.5$$ then $$\mathrm E \left(S_n\right) = 0.$$

For $$1\le i\lt j \le n$$, random variables $$Z_i$$ and $$Z_j$$ are independent, so we have $$\mathrm E \left(Z_i Z_j\right) = \mathrm E \left(Z_i\right) \mathrm E \left(Z_j\right).$$ We can now easily calculate variance of $$S_n:$$ \begin{aligned}\mathrm{Var}\left(S_n\right) &= \mathrm E \left(S^2_n\right) - \mathrm E ^2\left(S_n\right) \\ &= \mathrm E \left(\sum_{i=1}^n Z_i^2 + \sum_{1\le i\ne j\le n} Z_i Z_j \right) - \left(n\left(2p-1\right)\right)^2 \\ &= \sum_{i=1}^n \mathrm E \left(Z_i^2\right) + \sum_{1\le i\ne j\le n} \mathrm E \left(Z_i\right) \mathrm E \left(Z_j\right) - n^2\left(2p-1\right)^2 \\ &= n+2n\left(n-1\right)\left(2p-1\right)^2 - n^2\left(2p-1\right)^2 \\ &= 4np(1-p).\end{aligned} Specially, if $$p=0.5$$ then $$\mathrm{Var}\left(S_n\right) = n.$$

Source code available on Github.