Quiz -12

10/26/23, 4:04 PM Quiz -12 localhost:8888/nbconvert/html/Quiz -12.ipynb?download=false 1/15 In [10]: import pandas as pd import numpy as np # Setting a random seed for reproducibility np . random . seed ( 789 ) # Generating conversions for Page A (e.g., a 20% conversion rate) and Page B (e.g., a 25% conversion rate) page_a_results = np . random . binomial ( 1 , 0.20 , 3000 ) page_b_results = np . random . binomial ( 1 , 0.25 , 3000 ) # Creating the dataset data = pd . DataFrame ({ 'page' : [ 'A' ] * 3000 + [ 'B' ] * 3000 , 'converted' : np . concatenate ([ page_a_results , page_b_results ]) }) # Saving the dataset to a CSV file data . to_csv ( 'landing_page_conversion.csv' , index = False ) In [11]: data = pd . read_csv ( 'landing_page_conversion.csv' ) In [12]: data

10/26/23, 4:04 PM Quiz -12 localhost:8888/nbconvert/html/Quiz -12.ipynb?download=false 2/15 <class 'pandas.core.frame.DataFrame'> RangeIndex: 6000 entries, 0 to 5999 Data columns (total 2 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 page 6000 non-null object 1 converted 6000 non-null int64 dtypes: int64(1), object(1) memory usage: 93.9+ KB Out[12]: In [13]: data . info () In [14]: data . head ( 5 )

10/26/23, 4:04 PM Quiz -12 localhost:8888/nbconvert/html/Quiz -12.ipynb?download=false 4/15 Initial statistics of conversions for each page: count mean page A 3000 0.198333 B 3000 0.264333 Posterior distribution parameters for Page A: Alpha = 597 Beta = 2415 Posterior distribution parameters for Page B: Alpha = 795 Beta = 2217 In [22]: import matplotlib.pyplot as plt import scipy.stats as stats # Define the prior belief as a beta distribution prior_alpha = 2 prior_beta = 10 # Update the prior belief based on observed conversion rates for each page observed_page_a = data [ data [ 'page' ] == 'A' ][ 'converted' ] observed_page_b = data [ data [ 'page' ] == 'B' ][ 'converted' ] posterior_alpha_page_a = prior_alpha + sum ( observed_page_a ) posterior_beta_page_a = prior_beta + len ( observed_page_a ) - sum ( observed_page_a ) posterior_alpha_page_b = prior_alpha + sum ( observed_page_b ) posterior_beta_page_b = prior_beta + len ( observed_page_b ) - sum ( observed_page_b ) # Calculate the posterior distribution of the conversion rates for both pages using the beta distribution x = np . linspace ( 0 , 1 , 100 ) y_page_a = stats . beta . pdf ( x , posterior_alpha_page_a , posterior_beta_page_a ) y_page_b = stats . beta . pdf ( x , posterior_alpha_page_b , posterior_beta_page_b ) print ( "Posterior distribution parameters for Page A: Alpha =" , posterior_alpha_page_a , "Beta =" , posterior_be print ( "Posterior distribution parameters for Page B: Alpha =" , posterior_alpha_page_b , "Beta =" , posterior_be In [21]: # Define the prior belief as a beta distribution prior_alpha = 2 prior_beta = 10

10/26/23, 4:04 PM Quiz -12 localhost:8888/nbconvert/html/Quiz -12.ipynb?download=false 5/15 # Update the prior belief based on observed conversion rates for each page observed_page_a = data [ data [ 'page' ] == 'A' ][ 'converted' ] observed_page_b = data [ data [ 'page' ] == 'B' ][ 'converted' ] posterior_alpha_page_a = prior_alpha + sum ( observed_page_a ) posterior_beta_page_a = prior_beta + len ( observed_page_a ) - sum ( observed_page_a ) posterior_alpha_page_b = prior_alpha + sum ( observed_page_b ) posterior_beta_page_b = prior_beta + len ( observed_page_b ) - sum ( observed_page_b ) # Calculate the prior and posterior distributions of the conversion rates for both pages using the beta distri x = np . linspace ( 0 , 1 , 100 ) y_prior = stats . beta . pdf ( x , prior_alpha , prior_beta ) y_posterior_page_a = stats . beta . pdf ( x , posterior_alpha_page_a , posterior_beta_page_a ) y_posterior_page_b = stats . beta . pdf ( x , posterior_alpha_page_b , posterior_beta_page_b ) # Find the modes of the posterior distributions mode_page_a = ( posterior_alpha_page_a - 1 ) / ( posterior_alpha_page_a + posterior_beta_page_a - 2 ) mode_page_b = ( posterior_alpha_page_b - 1 ) / ( posterior_alpha_page_b + posterior_beta_page_b - 2 ) # Plot the prior and posterior distributions plt . figure ( figsize = ( 12 , 6 )) plt . plot ( x , y_prior , label = 'Prior Distribution' , color = 'g' , linestyle = '--' ) plt . plot ( x , y_posterior_page_a , label = 'Posterior Distribution Page A' , color = 'b' ) plt . plot ( x , y_posterior_page_b , label = 'Posterior Distribution Page B' , color = 'r' ) plt . axvline ( x = mode_page_a , color = 'b' , linestyle = '--' ) plt . axvline ( x = mode_page_b , color = 'r' , linestyle = '--' ) plt . text ( mode_page_a , 8 , f'Mode A: { mode_page_a :.2f}' , color = 'b' , ha = 'center' ) plt . text ( mode_page_b , 8 , f'Mode B: { mode_page_b :.2f}' , color = 'r' , ha = 'center' ) plt . title ( 'Prior and Posterior Distributions of Conversion Rates for Page A and Page B' ) plt . xlabel ( 'Conversion Rate' ) plt . ylabel ( 'Density' ) plt . legend () plt . show () print ( "Posterior distribution parameters for Page A: Alpha =" , posterior_alpha_page_a , "Beta =" , posterior_be print ( "Posterior distribution parameters for Page B: Alpha =" , posterior_alpha_page_b , "Beta =" , posterior_be

10/26/23, 4:04 PM Quiz -12 localhost:8888/nbconvert/html/Quiz -12.ipynb?download=false 7/15

10/26/23, 4:04 PM Quiz -12 localhost:8888/nbconvert/html/Quiz -12.ipynb?download=false 8/15 In [23]: import pandas as pd import numpy as np np . random . seed ( 456 ) def generate_transitions ( n ): states = [ 'S' , 'C' , 'D' ] # A mock transition matrix (you can adjust these values) transition_matrix = { 'S' : { 'S' : 0.7 , 'C' : 0.1 , 'D' : 0.2 }, 'C' : { 'S' : 0 , 'C' : 1 , 'D' : 0 }, 'D' : { 'S' : 0.4 , 'C' : 0.3 , 'D' : 0.3 } } current_state = np . random . choice ( states ) sequence = [ current_state ] for _ in range ( n - 1 ): next_state = np . random . choice ( list ( transition_matrix [ current_state ] . keys ()), p = list ( transition_matrix [ current_state ] . values ())) sequence . append ( next_state ) current_state = next_state return sequence

10/26/23, 4:04 PM Quiz -12 localhost:8888/nbconvert/html/Quiz -12.ipynb?download=false 10/15 Out[27]: In [28]: # Check for any missing values missing_values = data . isnull () . sum () # Handle missing values appropriately if missing_values . any (): data = data . dropna () In [29]: # Display the transition count for consecutive months transitions = data . apply ( lambda row : [( row [ i ], row [ i + 1 ]) for i in range ( len ( row ) - 1 )], axis = 1 ) transition_counts = pd . Series ( transitions . values . flatten ()) . value_counts () print ( "Transition Count for Consecutive Months:" ) print ( transition_counts )

10/26/23, 4:04 PM Quiz -12 localhost:8888/nbconvert/html/Quiz -12.ipynb?download=false 11/15 Transition Count for Consecutive Months: [(C, C), (C, C), (C, C), (C, C)] 335 [(D, C), (C, C), (C, C), (C, C)] 98 [(S, S), (S, S), (S, S), (S, S)] 70 [(D, S), (S, S), (S, S), (S, S)] 55 [(S, S), (S, S), (S, S), (S, D)] 33 [(D, D), (D, C), (C, C), (C, C)] 33 [(S, C), (C, C), (C, C), (C, C)] 31 [(S, D), (D, C), (C, C), (C, C)] 23 [(D, D), (D, S), (S, S), (S, S)] 20 [(S, S), (S, C), (C, C), (C, C)] 18 [(S, S), (S, S), (S, C), (C, C)] 18 [(S, S), (S, S), (S, S), (S, C)] 18 [(D, S), (S, S), (S, C), (C, C)] 17 [(D, S), (S, S), (S, S), (S, D)] 14 [(S, D), (D, S), (S, S), (S, S)] 13 [(S, S), (S, S), (S, D), (D, C)] 11 [(S, S), (S, S), (S, D), (D, S)] 11 [(D, S), (S, C), (C, C), (C, C)] 10 [(S, D), (D, D), (D, S), (S, S)] 10 [(S, S), (S, D), (D, D), (D, D)] 9 [(D, D), (D, D), (D, S), (S, S)] 9 [(D, S), (S, S), (S, D), (D, D)] 8 [(D, S), (S, D), (D, C), (C, C)] 8 [(S, S), (S, D), (D, C), (C, C)] 8 [(D, D), (D, D), (D, D), (D, C)] 7 [(D, S), (S, S), (S, S), (S, C)] 7 [(D, S), (S, S), (S, D), (D, C)] 7 [(D, S), (S, D), (D, S), (S, S)] 7 [(D, D), (D, S), (S, C), (C, C)] 6 [(S, S), (S, S), (S, D), (D, D)] 6 [(D, S), (S, S), (S, D), (D, S)] 6 [(D, D), (D, D), (D, C), (C, C)] 5 [(S, S), (S, D), (D, S), (S, S)] 5 [(S, D), (D, S), (S, S), (S, D)] 4 [(D, D), (D, S), (S, S), (S, D)] 4 [(D, D), (D, S), (S, D), (D, C)] 4 [(S, D), (D, D), (D, C), (C, C)] 4 [(S, D), (D, S), (S, D), (D, C)] 4 [(D, D), (D, D), (D, D), (D, D)] 4 [(D, D), (D, D), (D, S), (S, D)] 3 [(S, D), (D, D), (D, D), (D, D)] 3 [(D, D), (D, D), (D, D), (D, S)] 3 [(S, D), (D, S), (S, S), (S, C)] 3 [(S, D), (D, S), (S, D), (D, S)] 3

10/26/23, 4:04 PM Quiz -12 localhost:8888/nbconvert/html/Quiz -12.ipynb?download=false 13/15 # Visualize the transition probability matrix using a heatmap plt . figure ( figsize = ( 8 , 6 )) sns . heatmap ( transition_matrix , annot = True , cmap = 'YlGnBu' , cbar = True ) plt . title ( 'Transition Probability Matrix' ) plt . xlabel ( 'To State' ) plt . ylabel ( 'From State' ) plt . show ()

10/26/23, 4:04 PM Quiz -12 localhost:8888/nbconvert/html/Quiz -12.ipynb?download=false 14/15 Steady-State Probabilities: S: 1.5639809708684383e-14 C: 8.881512708122446e-13 D: 5.6104326048076704e-15 Predicted State Distribution after 10 Months: S: 0.03352152021388491 C: 0.0 D: 0.025435765813770654 In [33]: # Calculate the steady-state probabilities steady_state_probabilities = np . linalg . matrix_power ( transition_matrix , 100 ) steady_state_probabilities = steady_state_probabilities [ 0 , :] print ( "Steady-State Probabilities:" ) print ( "S:" , steady_state_probabilities [ 0 ]) print ( "C:" , steady_state_probabilities [ 1 ]) print ( "D:" , steady_state_probabilities [ 2 ]) # Predict the state distribution of the cohort after 10 months initial_state_distribution = [ 1 , 0 , 0 ] # Assuming all customers start in state 'S' predicted_state_distribution = np . dot ( np . linalg . matrix_power ( transition_matrix , 10 ), initial_state_distributi print ( "\nPredicted State Distribution after 10 Months:" ) print ( "S:" , predicted_state_distribution [ 0 ]) print ( "C:" , predicted_state_distribution [ 1 ]) print ( "D:" , predicted_state_distribution [ 2 ])