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Hints for Lab 6: Random walk and Brownian motion

Hints for Lab 6: Random walk and Brownian motion#

For the class on Wednesday, January 31st

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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!

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