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step has an equal probability of making a step of ±1. +Plot an ensemble of 10 trajectories, each 1000 points long, and overlay error bars of width 3sigma(t) on the plot. +''' +import numpy as np +import time + +def build_LFSR(seed, bits_needed=1000): + depth = 24 + taps = [1,2,7,24] + state = seed + output = [] + for _ in range(bits_needed): + newbit = (state ^ (state >> taps[1]) ^ (state >> taps[2]) ^ (state >> taps[3])) & 1 + state = (state >> 1) | (newbit << (depth - 1)) # shift everything over once and or our new bit in + output.append(newbit) + return output + +# Initialize a numpy array of zeros 10x1000 +walker_states = np.zeros((10, 1000)) + +# create our seeds by grabbing a 16 bit number from the LSB of the current time in us +# cheating and just adding a fixed delay to get different seeds +seeds = [] +for i in range(10): + delay = .1 + seeds.append(int(str(time.time())[-3:])) + time.sleep(delay) + +for i in range(10): + random_path = np.array(build_LFSR(seeds[i])) + random_path[(random_path == 0)] = -1 # turn our os to -1s so we can actually walk + for j in range(1000): + walker_states[i, j] = walker_states[i, j-1] + random_path[j] + +# Brownian motion is normally distributed with variance = t, so standard deviation will be sqrt(t), so let's get all of our timesteps now +timestep = np.array(range(1000)) + +import matplotlib.pyplot as plt +plt.plot(walker_states.T) # transpose it so we can plot it + +# for the 3 sigma lines, we will plot the sqrt of the timestep and divide the result by 1.5 to plot above and below the mean (0) +plt.plot(timestep, 1.5 * np.sqrt(timestep), color='black', alpha = 0.25) +plt.plot(timestep, -1.5 * np.sqrt(timestep), color='black', alpha = 0.25) +plt.show() + +''' +Part E: What fraction of the trajectories should be contained in the error bars? + +# We can refer to a standard normal distribution table (alternative to integrating) +Z-tables have the first two digits of Z-score in Y and the following 2 in X, so for 1.5 we look at the first entry of the +1.5 column +and the first entry of the +1.5 row, which is 0.93319. (from wikipedia) +https://en.wikipedia.org/wiki/Standard_normal_table + +Our actual answer is: +''' +probability_within = 0.93319 - (1 - 0.93319) +print(f"{probability_within*100:.2f} % to be within 3 sigma at any given time.") -- GitLab