Hints for Lab 6: Random walk and Brownian motion

For the class on Wednesday, January 31st

See also

Go back to Lab 6

A. Scaling and convergence of the diffusively rescaled random walk

Hints for Part A

• To calculate the quantile function (recall Lab 1A), you can use the following calc_quantile function.

  import numpy as np
  
  def calc_quantile(data):
      n = np.asanyarray(data).size
      q = (np.arange(n) + 0.5) / n
      return q, np.quantile(data, q)
  

• You can compare the calculated quantile function with any known quantile function by making a q-q plot. Below is an example of comparing with the Standard normal distribution.

  import scipy.stats
  import matplotlib.pyplot as plt
  
  q, qf = calc_quantile(data)
  plt.plot(qf, scipy.stats.norm.ppf(q))
  plt.xlabel("data")
  plt.ylabel("normal")
  

B. Universality of simple random walks

Hints for Part B

• Use np.random.default_rng().uniform to sample \(\mathcal{U}(-1, 1)\)

  s = np.random.default_rng().uniform(-1, 1, size=n_steps_total)
  

• The new distribution of \(W_{N=100}(t=1)\) may look like a normal distribution but with a different variance!