Mathematics and Statistics for the Quantitative Sciences

✍ Scribed by Matthew Betti

• Publisher: CRC Press/Chapman & Hall
• Year: 2022
• Tongue: English
• Leaves: 472
• Category: Library

✦ Synopsis

Mathematics and Statistics for the Quantitative Sciences was born from a radical reimagining of first-year mathematics. While calculus is often seen as the foundational mathematics required for any scientist, this often leads to mathematics being seen as some, ultimately useless, hoop that needs to be jumped through in order to do what someone really wants to do. This sentiment is everywhere at every level of education. It even shows up in how people stereotype mathematics courses.

What this book aims to do, therefore, is serve as a foundational text in everyday mathematics in a way that is both engaging and practically useful. The book seeks to teach the mathematics needed to start to answer fundamental questions like ‘why’ or ‘how’. Why do we only need to take census data once every few years? How do we determine the optimal dosing of a new pharmaceutical without killing people in the process? Or, more generally, what does it even mean to be average? Or what does it mean for two things to actually be different? These questions require a different way of thinking ― a quantitative intuition that goes beyond rote memorization and equips readers to meet the quantitative challenges inherent in any applied discipline.

Features

• Draws from a diverse range of fields to make the applications as inclusive as possible
• Would be ideal as a foundational mathematical and statistical textbook for any applied quantitative science course.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Author Bio
SECTION I: Applied Mathematics
The Plot (so you don’t lose it)
CHAPTER 1: Functions
1.1. ANATOMY OF A FUNCTION
1.2. MODELLING WITH MATHEMATICS
1.3. CONSTANTS AND LINEAR FUNCTIONS
1.4. POLYNOMIALS
1.5. EXPONENTIALS AND LOGARITHMS
1.6. FUNCTIONS IN HIGHER DIMENSIONS
1.7. CONTOUR DIAGRAMS
1.8. MODELS IN TWO DIMENSIONS
1.9. VARIABLES VS. PARAMETERS
CHAPTER 2: Derivatives
2.1. THE TANGENT LINE
2.2. APPROXIMATING DERIVATIVES OF FUNCTIONS
2.3. LIMITS
2.4. LIMITS AND DERIVATES
2.5. DERIVATIVE FORMULAS
2.6. THE PRODUCT RULE
2.7. THE CHAIN RULE
2.8. MIXING RULES
2.9. CRITICAL VALUES
2.10. CONSTRAINED OPTIMIZATION
2.11. ELASTICITY
2.12. PARTIAL DERIVATIVES
CHAPTER 3: Linear Algebra
3.1. VECTORS
3.2. MATRICES
3.3. MULTIPLICATION: NUMBERS AND MATRICES
3.4. MULTIPLICATION: MATRIX AND VECTORS
3.5. MULTIPLICATION: MATRIX AND MATRIX
3.6. LESLIE MATRICES
3.7. THE DETERMINANT
3.8. EIGENVALUES & EIGENVECTORS
CHAPTER 4: Derivatives in Multiple Dimensions
4.1. APPLICATIONS
4.2. DISTRIBUTION FITTING, PROBABILITY, AND LIKELIHOOD
CHAPTER 5: Differential Equations
5.1. SOLVING BASIC DIFFERENTIAL EQUATIONS: WITH AN EXAMPLE
5.2. EQUILIBRIA AND STABILITY
5.3. EQUILIBRIA AND LINEAR STABILITY IN HIGHER DIMENSIONS
5.4. THE JACOBIAN
CHAPTER 6: Integration
6.1. ACCUMULATED CHANGE
6.2. THE FUNDAMENTAL THEOREM OF CALCULUS
6.3. THE ANTI-DERIVATIVE
6.4. FUNDAMENTAL THEOREM OF CALCULUS REVISITED
6.5. PROPERTIES OF INTEGRALS
6.6. INTEGRATION BY PARTS
6.7. SUBSTITUTION
SECTION II: Applied Stats & Data Science
Some Context to Anchor Us
Math Versus The World
CHAPTER 7: Data and Summary Statistics
7.1. WHAT IS DATA?
7.2. DATA IN PYTHON
7.3. SUMMARY STATISTICS
7.4. ETHICAL AND MORAL CONSIDERATIONS: PART 1
7.5. MEAN VS. MEDIAN VS. MODE
7.6. VARIANCE AND STANDARD DEVIATION
7.7. ETHICAL AND MORAL CONSIDERATIONS: EPISODE 2
7.8. AN EXAMPLE
7.9. THE EMPIRICAL RULE
CHAPTER 8: Visualizing Data
8.1. PLOTTING IN PYTHON
8.2. SCATTER PLOTS
8.3. OUTLIERS
8.4. HISTOGRAMS
8.5. THE ANATOMY OF A TECHNICAL DOCUMENT
8.6. BAD PLOTS AND WHY THEY’RE BAD
CHAPTER 9: Probability
9.1. ETHICAL AND MORAL CONSIDERATIONS: A VERY SPECIAL EPISODE
9.2. COUNTING
9.3. PERMUTATIONS
9.4. COMBINATIONS
9.5. COMBINATIONS WITH REPLACEMENT
9.6. PROBABILITY
9.7. PROPERTIES OF PROBABILITIES
9.8. MORE NOTATION
9.9. CONDITIONAL PROBABILITY
9.10. BAYES’ THEOREM
9.11. THE PROSECUTOR’S FALLACY
9.12. THE LAW OF TOTAL PROBABILITY
CHAPTER 10: Probability Distributions
10.1. DISCRETE PROBABILITY DISTRIBUTIONS
10.2. THE BINOMIAL DISTRIBUTION
10.3. TRINOMIAL DISTRIBUTION
10.4. CUMULATIVE PROBABILITY DISTRIBUTIONS
10.5. CONTINUOUS PROBABILITY
10.6. CONTINUOUS VS. DISCRETE PROBABILITY DISTRIBUTIONS
10.7. PROBABILITY DENSITY FUNCTIONS
10.8. THE NORMAL DISTRIBUTION
10.9. OTHER USEFUL DISTRIBUTIONS
10.10. MEAN, MEDIAN, MODE, AND VARIANCE
10.11. SUMMING TO INFINITY
10.12. PROBABILITY AND PYTHON
10.13. PRACTICE PROBLEMS
CHAPTER 11: Fitting Data
11.1. DEFINING RELATIONSHIPS
11.2. DATA AND LINES
11.3. DISTRIBUTION FITTING AND LIKELIHOOD
11.4. DUMMY VARIABLES
11.5. LOGISTIC REGRESSION
11.6. LOGISTIC REGRESSION IN PYTHON
11.7. ITERATED LOGISTIC REGRESSION
11.8. RANDOM FOREST CLASSIFICATION
11.9. BOOTSTRAPPING AND CONFIDENCE INTERVALS
11.10. T-STATISTICS
11.11. THE DICHOTOMOUS NATURE OF P-VALUES
APPENDIX A: A Crash Course in Python
A.I. VARIABLES
A.II. KEYWORDS
A.III. CONDITIONALS
A.IV. LOOPS
A.V. IMPORT
A.VI. FUNCTIONS
A.VII. A SIMPLE PYTHON PROGRAM
Bibliography
Index