import matplotlib.pyplot as plt
   import numpy as np
   import pandas as pd
   num_realizations = 300
   len_process = 1000
   input_std = 0.1
   random_nrs = pd.DataFrame(
       np.random.normal(loc=0, scale=input_std, size=(num_realizations, len_process))
   )
   # random_nrs.cumsum(axis=1).T.plot()
   random_nrs.cumsum(axis=1).var(axis=0).plot(
       label=r"estimated $\hat{Var}\left (\sum_{i=1}^N x_i \right)$"
   )
   input_var = input_std**2
   plt.plot(
       np.arange(len_process) * input_var,
       label=r"expected $Var\left (\sum_{i=1}^N x_i \right)=N \sigma^2$",
   )
   plt.xlabel("N")
   plt.title(f"Averages are taken over {num_realizations} realizations")
   plt.ylabel(r"Var$\left (\sum_{i=1}^N x_i \right)$")
   plt.legend()