Assignment 12 - Problems

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School

Northeastern University *

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Course

6400

Subject

Industrial Engineering

Date

Feb 20, 2024

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html

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5

Uploaded by CaptainSparrowMaster1000

IE6400 Foundations for Data Analytics Engineering Fall 2023 Assignment 12 Module 2: Probability Question 1: Monty Hall Problem Simulation and Analysis Background: The Monty Hall problem is a famous probability puzzle named after the host of the television game show "Let's Make a Deal." The problem goes as follows: • A contestant is presented with three doors. Behind one of them is a car (which the contestant wants), and behind the other two are goats. • The contestant selects one of the doors, say Door A. • The host, Monty Hall, who knows what's behind each door, opens another door, say Door B, revealing a goat. • Monty now asks the contestant if they want to stick with their initial choice (Door A) or switch to the remaining door (Door C). • The contestant makes a decision: stick or switch. • The question is, is it in the contestant's best interest to stick with their initial choice, switch, or does it not matter? Task: Your goal is to simulate the Monty Hall problem using Python and determine the empirical probabilities of winning the car for both strategies: sticking with the initial choice and switching after Monty reveals a goat. Dataset: 'monty_hall_trials.csv' In [ ]: import pandas as pd import numpy as np np.random.seed(999) def simulate_monty_hall(num_trials): doors = ['A', 'B', 'C'] results = [] for i in range(num_trials): car_location = np.random.choice(doors) initial_choice = np.random.choice(doors) remaining_doors = [door for door in doors if door != initial_choice and door != car_location] monty_reveal = np.random.choice(remaining_doors) # Simulating equal probability of sticking or switching final_decision = np.random.choice(['Stick', 'Switch']) if final_decision == 'Stick': win = 1 if initial_choice == car_location else 0 else: switch_to = [door for door in doors if door != initial_choice and door != monty_reveal][0]
win = 1 if switch_to == car_location else 0 results.append([i+1, initial_choice, monty_reveal, car_location, final_decision, win]) return pd.DataFrame(results, columns=['trial', 'initial_choice', 'monty_reveal', 'actual_car_location', 'final_decision', 'win']) df = simulate_monty_hall(1000) df.to_csv('monty_hall_trials.csv', index=False) The dataset contains six columns: • trial: The trial number. • initial_choice: The initial door chosen by the contestant. • monty_reveal: The door Monty reveals to have a goat. • actual_car_location: The door behind which the car is actually located. • final_decision: The contestant's final decision, either "Stick" or "Switch". • win: Whether the contestant won the car (1 for win, 0 for lose). Requirements: 1. Data Loading and Preprocessing: • Load the dataset monty_hall_trials.csv into a Pandas DataFrame. • Check for any missing or inconsistent data entries and handle them. • Display a summary of the dataset. 2. Simulation Analysis: • Calculate the empirical probability of winning the car for both strategies: sticking and switching. 3. Visualization: • Plot a bar chart comparing the winning probabilities for both strategies. • Ensure the graph is appropriately labeled with a relevant title and annotations. 4. Interpretation: • Discuss the empirical results in the context of the theoretical probabilities. • Offer insights into the optimal strategy for a contestant based on the simulation results. Evaluation Criteria: • Correctness and efficiency of the Python code. • Proper handling and preprocessing of the dataset. • Accurate calculation and interpretation of empirical probabilities. • Quality and clarity of visualizations. • Insightful interpretations and conclusions regarding the Monty Hall problem. Question 2: Poisson Process Analysis of Website Hits Background: A Poisson process is a mathematical model for events that happen at random points in time and space, where the average rate of occurrence is constant and known. A common application of this process is in modeling the number of times a website is accessed over a given time interval. Scenario: You are a data analyst at a tech company. The company's main website has been receiving hits, and you suspect that the hits can be modeled as a Poisson process. Your task is to analyze the website hits data and verify if it indeed follows the Poisson distribution. Dataset: 'website_hits.csv' In [ ]: import pandas as pd import numpy as np
np.random.seed(12345) # Generating hits using Poisson distribution # Assuming mean hits per hour is 6 hits_per_hour = np.random.poisson(lam=6, size=24) time_intervals = [f"{i}-{i+1}" for i in range(24)] df = pd.DataFrame({ 'time_interval': time_intervals, 'hits': hits_per_hour }) df.to_csv('website_hits.csv', index=False) The dataset contains two columns: • time_interval: Represents hourly intervals over a 24-hour period (e.g., "0-1" represents the interval from midnight to 1 AM). • hits: The number of website hits recorded during the corresponding time interval. Requirements: 1. Data Loading and Preprocessing: • Load the dataset website_hits.csv into a Pandas DataFrame. • Check for any missing or inconsistent data entries and handle them. • Display the basic statistics of the dataset. 2. Poisson Distribution Fitting: • Calculate the mean hit rate from the data. • Using the calculated mean, generate the expected hit frequencies for each hour if the process follows a Poisson distribution. 3. Visualization: • Plot a bar chart showing both the observed and expected hits for each hourly interval. The bars for observed and expected hits should be side-by-side for comparison. • Ensure the graph is properly labeled with a relevant title, legend, and annotations. 4. Hypothesis Testing: • Conduct a goodness-of-fit test (e.g., chi-squared test) to determine if the observed hits significantly differ from a Poisson distribution with the calculated mean rate. 5. Interpretation: • Discuss the results of the visualization and hypothesis test. • Provide insights and recommendations to the company based on your findings. Evaluation Criteria: • Correctness and efficiency of the Python code. • Proper handling and preprocessing of the dataset. • Accurate fitting of the Poisson distribution and calculation of expected frequencies. • Quality and clarity of visualizations. • Thoroughness of hypothesis testing. • Insightful interpretations and conclusions drawn from the analysis. Question 3: Bayesian Analysis of Product Review Sentiments Background: Bayesian statistics is a branch of probability theory that applies probability to
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