SS Exercises

import math import matplotlib.pyplot as plt import numpy as np import statistics import random import pandas as pd import sys

def geometric_distribution(p, n): return pow((1-p), n-1)*p

#set the probability parameter p = 1/6 probabilities = [] #create list with 100 samples from geometric distribution for n in range(1,100): probabilities.append(geometric_distribution(p,n))

plt.figure(figsize=(16,8)) plt.bar(range(1,100), probabilities) plt.xlabel('n-value') plt.title('Geometric Probability Distribution') plt.show()

p = 1/6 probabilities = [] #create 10.000 samples for n in range(1,10000): probabilities.append(geometric_distribution(p,n)) print("Simulate 10.000 outcomes for Geometric distribution") print(probabilities)

X = [1, 2, 3, 4, 5, 6] p_i = [7/48, 5/48, 1/8, 1/16, 1/4, 5/16] direct_method = [] #simulate 1000 samples based on probabilities given for i in range(0,1000): n = np.random.choice(np.arange(1, 7), p=p_i) direct_method.append(n)

plt.figure(figsize=(16,8)) norm_count_dir = pd.DataFrame({'Number': direct_method}).value_counts(normalize=True, sort=False) norm_count_dir.plot(kind='bar') plt.title('Direct (crude) Method Random Number Sample Split') plt.show()

c = max(p_i) k = 6 samples = [] #simulate 1000 samples using rejection method for i in range(0, 1000): success = False while(success is not True): I = int((k*random.random()+1)) if(random.random() <= p_i[I-1]/c): samples.append(I) success = True print(samples)

plt.figure(figsize=(16,8)) norm_count_rej = pd.DataFrame({'Number': samples}).value_counts(normalize=True, sort=False) norm_count_rej.plot(kind='bar') plt.title('Rejection Method Random Number Sample Split') plt.show()

k = len(p_i) L = np.zeros(k, dtype=np.int) F = np.zeros(k, dtype=np.float) S = [] G = [] for kk, prob in enumerate(p_i): F[kk] = k*prob if F[kk] < 1.0: S.append(kk) else: G.append(kk)

while len(G) > 0 and len(S) > 0: small = S.pop() large = G.pop() L[small] = large F[large] = F[large] - (1-F[small]) if F[large] < 1.0: S.append(large) else: G.append(large)

k = 6 samples = [] for i in range(0, 1000): I = int((k*random.random())) if(random.random() <= F[I]): samples.append(I+1) else: samples.append(L[I]+1) print(samples)

norm_count_alias = pd.DataFrame({'Number': samples}).value_counts(normalize=True, sort=False) print("Dataframe with samples created using Alias method") print(norm_count_alias)

x = [1,2,3,4,5,6] plt.figure(figsize=(16,8)) plt.plot(x, norm_count_alias.tolist(), label='Alias') plt.plot(x, norm_count_rej.tolist(), label='Rejection') plt.plot(x, norm_count_dir.tolist(), label='Direct') plt.plot(x, p_i, label='Original Probability') plt.title('Alias Method Random Number Sample Split') plt.legend() plt.show()

alias_difference = [abs(x-y) for x,y in zip(p_i,norm_count_alias.tolist())] rej_difference = [abs(x-y) for x,y in zip(p_i,norm_count_rej.tolist())] dir_difference = [abs(x-y) for x,y in zip(p_i,norm_count_dir.tolist())] print("Difference between original probability and Alias method",sum(alias_difference)) print("Difference between original probability and rejection method",sum(rej_difference)) print("Difference between original probability and Direct method",sum(dir_difference))