The Statistics and Calculus with Python Workshop: A comprehensive introduction to mathematics in Python for artificial intelligence applications

The Statistics and Calculus with Python Workshop....2 Preface....20 About the Book....20 Audience....20 About the Chapters....20 Conventions....21 Code Presentation....22 Setting up Your Environment....22 Software Requirements....22 Installation and Setup....22 Installing Python....23 Project Jupyter....23 Installing Libraries....25 Accessing the Code Files....25 1. Fundamentals of Python....27 Introduction....27 Control Flow Methods....27 if Statements....27 Exercise 1.01: Divisibility with Conditionals....28 Loops....30 The while Loop....30 The for Loop....30 Exercise 1.02: Number Guessing Game....31 Data Structures....34 Strings....34 Lists....35 Exercise 1.03: Multi-Dimensional Lists....36 Tuples....38 Sets....38 Dictionaries....39 Exercise 1.04: Shopping Cart Calculations....40 Functions and Algorithms....42 Functions....42 Exercise 1.05: Finding the Maximum....44 Recursion....45 Exercise 1.06: The Tower of Hanoi....46 Algorithm Design....47 Exercise 1.07: The N-Queens Problem....47 Testing, Debugging, and Version Control....51 Testing....51 Debugging....52 Exercise 1.08: Testing for Concurrency....53 Version Control....57 Exercise 1.09: Version Control with Git and GitHub....58 Activity 1.01: Building a Sudoku Solver....61 Summary....63 2. Python's Main Tools for Statistics....64 Introduction....64 Scientific Computing and NumPy Basics....64 NumPy Arrays....65 Vectorization....69 Exercise 2.01: Timing Vectorized Operations in NumPy....69 Random Sampling....71 Working with Tabular Data in pandas....74 Initializing a DataFrame Object....74 Accessing Rows and Columns....75 Manipulating DataFrames....78 Exercise 2.02: Data Table Manipulation....78 Advanced Pandas Functionalities....82 Exercise 2.03: The Student Dataset....84 Data Visualization with Matplotlib and Seaborn....87 Scatter Plots....88 Line Graphs....89 Bar Graphs....91 Histograms....94 Heatmaps....96 Exercise 2.04: Visualization of Probability Distributions....96 Visualization Shorthand from Seaborn and Pandas....98 Activity 2.01: Analyzing the Communities and Crime Dataset....100 Summary....101 3. Python's Statistical Toolbox....103 Introduction....103 An Overview of Statistics....103 Types of Data in Statistics....104 Categorical Data....104 Exercise 3.01: Visualizing Weather Percentages....107 Numerical Data....109 Exercise 3.02: Min-Max Scaling....112 Ordinal Data....114 Descriptive Statistics....115 Central Tendency....115 Dispersion....116 Exercise 3.03: Visualizing Probability Density Functions....116 Python-Related Descriptive Statistics....118 Inferential Statistics....121 T-Tests....122 Correlation Matrix....124 Exercise 3.04: Identifying and Testing Equality of Means....125 Statistical and Machine Learning Models....127 Exercise 3.05: Model Selection....129 Python's Other Statistics Tools....132 Activity 3.01: Revisiting the Communities and Crimes Dataset....133 Summary....134 4. Functions and Algebra with Python....135 Introduction....135 Functions....135 Common Functions....136 Domain and Range....138 Function Roots and Equations....138 The Plot of a Function....138 Exercise 4.01: Function Identification from Plots....140 Function Transformations....141 Shifts....141 Scaling....142 Exercise 4.02: Function Transformation Identification....143 Equations....145 Algebraic Manipulations....145 Factoring....146 Using Python....148 Exercise 4.03: Introduction to Break-Even Analysis....149 Systems of Equations....151 Systems of Linear Equations....152 Exercise 4.04: Matrix Solution with NumPy....154 Systems of Non-Linear Equations....157 Activity 4.01: Multi-Variable Break-Even Analysis....159 Summary....160 5. More Mathematics with Python....161 Introduction....161 Sequences and Series....161 Arithmetic Sequences....162 Generators....164 Exercise 5.01: Determining the nth Term of an Arithmetic Sequence and Arithmetic Series....166 Geometric Sequences....168 Exercise 5.02: Writing a Function to Find the Next Term of the Sequence....169 Recursive Sequences....171 Exercise 5.03: Creating a Custom Recursive Sequence....172 Trigonometry....174 Basic Trigonometric Functions....174 Exercise 5.04: Plotting a Right-Angled Triangle....175 Inverse Trigonometric Functions....177 Exercise 5.05: Finding the Shortest Way to the Treasure Using Inverse Trigonometric Functions....179 Exercise 5.06: Finding the Optimal Distance from an Object....180 Vectors....182 Vector Operations....182 Exercise 5.07: Visualizing Vectors....185 Complex Numbers....188 Basic Definitions of Complex Numbers....189 Polar Representation and Euler's Formula....191 Exercise 5.08: Conditional Multiplication of Complex Numbers....194 Activity 5.01: Calculating Your Retirement Plan Using Series....196 Summary....196 6. Matrices and Markov Chains with Python....198 Introduction....198 Matrix Operations on a Single Matrix....198 Basic Operations on a Matrix....199 Inspecting a Matrix....201 Exercise 6.01: Calculating the Time Taken for Sunlight to Reach Earth Each Day....202 Operations and Multiplication in Matrices....205 Axes in a Matrix....208 Exercise 6.02: Matrix Search....209 Multiple Matrices....211 Broadcasting....212 Operations on Multiple Matrices....213 Identity Matrix....213 The eye Function....213 Inverse of a Matrix....214 Logical Operators....214 Outer Function or Vector Product....215 Solving Linear Equations Using Matrices....216 Exercise 6.03: Use of Matrices in Performing Linear Equations....217 Transition Matrix and Markov Chains....219 Fundamentals of Markov Chains....220 Stochastic versus Deterministic Models....220 Transition State Diagrams....220 Transition Matrices....225 Exercise 6.04: Finding the Probability of State Transitions....226 Markov Chains and Markov Property....229 Activity 6.01: Building a Text Predictor Using a Markov Chain....230 Summary....231 7. Doing Basic Statistics with Python....232 Introduction....232 Data Preparation....232 Introducing the Dataset....232 Introducing the Business Problem....233 Preparing the Dataset....234 Exercise 7.01: Using a String Column to Produce a Numerical Column....239 Calculating and Using Descriptive Statistics....240 The Need for Descriptive Statistics....240 A Brief Refresher of Statistical Concepts....241 Using Descriptive Statistics....245 Exercise 7.02: Calculating Descriptive Statistics....247 Exploratory Data Analysis....248 What Is EDA?....249 Univariate EDA....250 Bi-variate EDA: Exploring Relationships Between Variables....255 Exercise 7.03: Practicing EDA....257 Activity 7.01: Finding Out Highly Rated Strategy Games....259 Summary....260 8. Foundational Probability Concepts and Their Applications....261 Introduction....261 Randomness, Probability, and Random Variables....261 Randomness and Probability....262 Foundational Probability Concepts....262 Introduction to Simulations with NumPy....263 Exercise 8.01: Sampling with and without Replacement....266 Probability as a Relative Frequency....267 Defining Random Variables....270 Exercise 8.02: Calculating the Average Wins in Roulette....274 Discrete Random Variables....276 Defining Discrete Random Variables....276 The Binomial Distribution....278 Exercise 8.03: Checking If a Random Variable Follows a Binomial Distribution....280 Continuous Random Variables....282 Defining Continuous Random Variables....282 The Normal Distribution....284 Some Properties of the Normal Distribution....287 Exercise 8.04: Using the Normal Distribution in Education....291 Activity 8.01: Using the Normal Distribution in Finance....293 Summary....294 9. Intermediate Statistics with Python....295 Introduction....295 Law of Large Numbers....295 Python and Random Numbers....296 Exercise 9.01: The Law of Large Numbers in Action....296 Exercise 9.02: Coin Flipping Average over Time....297 A Practical Application of the Law of Large Numbers Seen in the Real World....299 Exercise 9.03: Calculating the Average Winnings for a Game of Roulette If We Constantly Bet on Red....300 Central Limit Theorem....303 Normal Distribution and the CLT....304 Random Sampling from a Uniform Distribution....304 Exercise 9.04: Showing the Sample Mean for a Uniform Distribution....304 Random Sampling from an Exponential Distribution....306 Exercise 9.05: Taking a Sample from an Exponential Distribution....307 Confidence Intervals....310 Calculating the Confidence Interval of a Sample Mean....310 Exercise 9.06: Finding the Confidence Interval of Polling Figures....313 Small Sample Confidence Interval....314 Confidence Interval for a Proportion....315 Hypothesis Testing....316 Parts of a Hypothesis Test....316 The Z-Test....317 Exercise 9.07: The Z-Test in Action....318 Proportional Z-Test....320 The T-Test....321 Exercise 9.08: The T-Test....322 2-Sample T-Test or A/B Testing....325 Exercise 9.09: A/B Testing Example....326 Introduction to Linear Regression....327 Exercise 9.10: Linear Regression....328 Activity 9.01: Standardized Test Performance....330 Summary....331 10. Foundational Calculus with Python....332 Introduction....332 Writing the Derivative Function....332 Exercise 10.01: Finding the Derivatives of Other Functions....334 Finding the Equation of the Tangent Line....335 Calculating Integrals....336 Using Trapezoids....339 Exercise 10.02: Finding the Area Under a Curve....339 Using Integrals to Solve Applied Problems....340 Exercise 10.03: Finding the Volume of a Solid of Revolution....341 Using Derivatives to Solve Optimization Problems....343 Exercise 10.04: Find the Quickest Route....344 Exercise 10.05: The Box Problem....345 Exercise 10.06: The Optimal Can....346 Exercise 10.07: Calculating the Distance between Two Moving Ships....346 Activity 10.01: Maximum Circle-to-Cone Volume....347 Summary....348 11. More Calculus with Python....349 Introduction....349 Length of a Curve....349 Exercise 11.01: Finding the Length of a Curve....352 Exercise 11.02: Finding the Length of a Sine Wave....353 Length of a Spiral....353 Exercise 11.03: Finding the Length of the Polar Spiral Curve....355 Exercise 11.04: Finding the Length of Insulation in a Roll....356 Exercise 11.05: Finding the Length of an Archimedean Spiral....356 Area of a Surface....357 The Formulas....357 Exercise 11.06: Finding the Area of a 3D Surface – Part 1....361 Exercise 11.07: Finding the Area of a 3D Surface – Part 2....362 Exercise 11.08: Finding the Area of a Surface – Part 3....363 Infinite Series....363 Polynomial Functions....363 Series....364 Convergence....365 Exercise 11.09: Calculating 10 Correct Digits of π....366 Exercise 11.10: Calculating the Value of π Using Euler's Expression....367 A 20th Century Formula....367 Interval of Convergence....368 Exercise 11.11: Determining the Interval of Convergence – Part 1....369 Exercise 11.12: Determining the Interval of Convergence – Part 2....371 Exercise 11.13: Finding the Constant....372 Activity 11.01: Finding the Minimum of a Surface....372 Summary....373 12. Intermediate Calculus with Python....375 Introduction....375 Differential Equations....375 Interest Calculations....376 Exercise 12.01: Calculating Interest....376 Exercise 12.02: Calculating Compound Interest – Part 1....378 Exercise 12.03: Calculating Compound Interest – Part 2....380 Exercise 12.04: Calculating Compound Interest – Part 3....381 Exercise 12.05: Becoming a Millionaire....381 Population Growth....383 Exercise 12.06: Calculating the Population Growth Rate – Part 1....384 Exercise 12.07: Calculating the Population Growth Rate – Part 2....386 Half-Life of Radioactive Materials....386 Exercise 12.08: Measuring Radioactive Decay....386 Exercise 12.09: Measuring the Age of a Historical Artifact....387 Newton's Law of Cooling....389 Exercise 12.10: Calculating the Time of Death....389 Exercise 12.11: Calculating the Rate of Change in Temperature....391 Mixture Problems....392 Exercise 12.12: Solving Mixture Problems – Part 1....393 Exercise 12.13: Solving Mixture Problems – Part 2....395 Exercise 12.14: Solving Mixture Problems – Part 3....397 Exercise 12.15: Solving Mixture Problems – Part 4....398 Euler's Method....398 Exercise 12.16: Solving Differential Equations with Euler's Method....398 Exercise 12.17: Using Euler's Method to Evaluate a Function....400 Runge-Kutta Method....401 Exercise 12.18: Implementing the Runge-Kutta Method....402 Pursuit Curves....403 Exercise 12.19: Finding Where the Predator Catches the Prey....403 Exercise 12.20: Using Turtles to Visualize Pursuit Curves....405 Position, Velocity, and Acceleration....406 Exercise 12.21: Calculating the Height of a Projectile above the Ground....406 An Example of Calculating the Height of a Projectile with Air Resistance....409 Exercise 12.22: Calculating the Terminal Velocity....411 Activity 12.01: Finding the Velocity and Location of a Particle....412 Summary....413 Appendix....414 1. Fundamentals of Python....414 Activity 1.01: Building a Sudoku Solver....414 2. Python's Main Tools for Statistics....418 Activity 2.01: Analyzing the Communities and Crime Dataset....418 3. Python's Statistical Toolbox....423 Activity 3.01: Revisiting the Communities and Crimes Dataset....423 4. Functions and Algebra with Python....427 Activity 4.01: Multi-Variable Break-Even Analysis....427 5. More Mathematics with Python....432 Activity 5.01: Calculating Your Retirement Plan Using Series....432 6. Matrices and Markov Chains with Python....435 Activity 6.01: Building a Text Predictor Using a Markov Chain....436 7. Doing Basic Statistics with Python....438 Activity 7.01: Finding Out Highly Rated Strategy Games....438 8. Foundational Probability Concepts and Their Applications....440 Activity 8.01: Using the Normal Distribution in Finance....441 9. Intermediate Statistics with Python....443 Activity 9.01: Standardized Test Performance....444 10. Foundational Calculus with Python....448 Activity 10.01: Maximum Circle-to-Cone Volume....448 11. More Calculus with Python....449 Activity 11.01: Finding the Minimum of a Surface....449 12. Intermediate Calculus with Python....453 Activity 12.01: Finding the Velocity and Location of a Particle....453