Mathematics and Statistics for Science

Description

Mathematics and statistics are the bedrock of modern science. No matter which branch of science you plan to work in, you simply cannot avoid quantitative approaches. And while you won’t always need to know a great deal of theory, you will need to know how to apply mathematical and statistical methods in realistic scenarios. That is precisely what this book teaches. It covers the mathematical and statistical topics that are ubiquitous in early undergraduate courses, but does so in a way that is directly linked to science. Beginning with the use of units and functions, this book covers key topics such as complex numbers, vectors and matrices, differentiation (both single and multivariable), integration, elementary differential equations, probability, random variables, inference and linear regression. Each topic is illustrated with widely-used scientific equations (such as the ideal gas law or the Nernst equation) and real scientific data, often taken directly from recent scientific papers. The emphasis throughout is on practical solutions, including the use of computational tools (such as Wolfram Alpha or R), not theoretical development. There is a large number of exercises, divided into mathematical drills and scientific applications, and full solutions to all the exercises are available to instructors. Mathematics and Statistics for Science covers the core methods in mathematics and statistics necessary for a university degree in science, highlighting practical solutions and scientific applications. Its pragmatic approach is ideal for students who need to apply mathematics and statistics in a real scientific setting, whether in the physical sciences, life sciences or medicine. Preface Acknowledgements Contents Part I Units and Measurement Chapter 1 Units 1.1 Numbers 1.2 Decimals 1.3 Orders of magnitude and scientific notation 1.4 Numbers and units 1.4.1 Moles, molar, litres and bars 1.5 Units in equations 1.5.1 The units on each side of an equation have to match 1.6 Unit conversion 1.7 Parts per million, parts per billion Exercises Revision Math drills for practice Scientific applications Chapter 2 Measurement, rounding and uncertainty 2.1 Precision and accuracy 2.2 Significant figures and rounding 2.2.1 Sequential rounding is not allowed 2.2.2 Rounding and scientific notation 2.3 Measurement uncertainty 2.4 Significant figures in equations 2.4.1 Round only at the end of a calculation 2.5 Uncertainty analysis 2.5.1 Adding or subtracting measurements 2.5.2 Multiplying or dividing measurements 2.5.3 A more complex example Exercises Math drills for practice Scientific applications Part II Functions and Complex Numbers Chapter 3 Functions 3.1 What is a function? 3.2 Domain of a function 3.2.1 The mathematical domain and range 3.2.2 The scientific domain and range 3.3 Graphing functions 3.4 Functions represented by a table 3.5 Functions and units 3.6 Proportionality 3.7 Piecewise-defined functions 3.8 Operations on functions 3.9 Function composition Exercises Revision Math drills for practice Scientific applications Functions Chapter 4 Exponential and log functions 4.1 Exponential functions 4.1.1 The logistic equation 4.1.2 The Gaussian function, or normal distribution 4.1.3 The Arrhenius equation 4.2 Log functions 4.2.1 The Nernst equation 4.2.2 Domain of a log function 4.2.3 Converting between different bases Exercises Math drills for practice Scientific applications Chapter 5 Periodic functions 5.1 Trigonometric functions 5.1.1 Trig functions and the unit circle 5.1.2 The sine function 5.1.3 The cosine function 5.1.4 The tangent function 5.2 Arcs and sectors 5.3 Trigonometric identities 5.4 Calculating period and frequency 5.5 Solving simple trigonometric equations 5.6 Polar coordinates 5.6.1 Graphs in polar coordinates 5.7 Periodic functions as models of the real world Exercises Math drills for practice Scientific applications Chapter 6 Linearising functions 6.1 Revision: the equation of a line 6.2 Lineweaver–Burke plots 6.3 Linearising the Arrhenius equation 6.4 Power laws Exercises Scientific applications Chapter 7 Complex numbers 7.1 The number i and other complex numbers 7.2 Adding and subtracting complex numbers 7.3 Multiplying complex numbers 7.4 Dividing complex numbers; the conjugate 7.5 The complex plane 7.5.1 Polar coordinates 7.6 Complex roots of polynomials 7.7 Euler’s formula 7.7.1 Multiplication and division in exponential notation Exercises Math drills for practice Scientific applications Part III Vectors, Matrices and Linear Systems Chapter 8 Vectors 8.1 Adding and subtracting vectors 8.2 Scalar multiplication 8.3 Parallel vectors 8.4 Length of a vector 8.5 Distance between two vectors 8.6 Unit vectors 8.6.1 Standard unit vectors 8.7 The angle between two vectors 8.7.1 The dot product 8.7.2 Computing the angle between two vectors 8.7.3 Orthogonal vectors Exercises Math drills for practice Scientific applications Chapter 9 Matrices 9.1 Some basic matrix properties 9.1.1 The entries of a matrix 9.1.2 Square matrices 9.1.3 Equal matrices 9.1.4 Adding and subtracting two matrices 9.1.5 Scalar multiplication 9.1.6 Zero matrices 9.2 Row and column vectors 9.3 Matrix multiplication 9.3.1 Multiplying 2 X 2 matrices 9.3.2 Multiplying larger matrices 9.3.3 Matrix multiplication is associative but not commutative 9.4 A matrix as a linear transformation 9.4.1 Transforming polygons 9.5 The inverse of a matrix 9.5.1 Identity matrices 9.5.2 Invertible square matrices 9.5.3 Inverses of larger matrices Exercises Math drills for practice Scientific applications Chapter 10 Systems of linear equations 10.1 Linear equations 10.2 Solutions in two dimensions 10.3 Solutions in three dimensions 10.3.1 What happens if there are more or fewer than three equations? Exercises Math drills for practice Scientific applications Chapter 11 Solving systems of linear equations using matrices 11.1 Writing linear systems as matrix equations 11.1.1 The general form of a linear system 11.1.2 Using the matrix inverse to solve a linear system 11.1.3 If the matrix A is invertible there is a unique solution 11.2 Computing matrix inverses 11.2.1 Wolfram Alpha 11.2.2 R 11.3 Warning: not all matrices have an inverse 11.3.1 The case of no solutions 11.3.2 The case of an infinite number of solutions 11.3.3 Stoichiometric problems have an infinite number of solutions 11.3.4 The PageRank problem has an infinite number of solutions 11.3.5 Non-square systems 11.4 Linear systems can be written in an alternative way Exercises Math drills for practice Scientific applications Part IV Differentiation: Functions of One Variable Chapter 12 Limits 12.1 The simple cases 12.1.1 Limits at infinity Example 12.2 Limits of ratios of functions 12.2.1 The case of 0/0 12.3 Horizontal asymptotes 12.3.1 Horizontal asymptotes of rational functions 12.4 Vertical asymptotes Exercises Math drills for practice Scientific applications Chapter 13 Differentiation as a limit 13.1 Motivating the definition 13.2 Distance and velocity 13.3 Rate of a chemical reaction Exercises Math drills for practice Scientific applications Chapter 14 Differentiation in practice 14.1 Differentiating polynomials 14.2 Differentiating trig functions 14.2.1 Using a computer or looking it up online 14.3 Differentiating exponential and log functions 14.4 Higher derivatives 14.5 Product rule 14.6 Quotient rule 14.7 Chain rule 14.7.1 Multiple steps of the chain rule 14.8 Using a computer 14.9 Positions and velocities; the derivative of a vector Exercises Math drills for practice Scientific applications Chapter 15 Numerical differentiation 15.1 Calculating approximate derivatives 15.2 Data with increased resolution 15.3 Problems with high resolution Exercises Scientific applications Chapter 16 Implicit differentiation 16.1 Using the chain rule 16.2 Relative rates of change Exercises Math drills for practice Scientific applications Chapter 17 Maxima and minima 17.1 Local maxima and local minima 17.2 Critical points and stationary points 17.3 Concavity and points of inflection 17.4 Second derivative test 17.4.1 The second derivative test in practice Exercises Math drills for practice Scientific applications Part V Differentiation: Functions of Multiple Variables Chapter 18 Functions of multiple variables 18.1 Graphing functions of two variables 18.1.1 The two-dimensional Gaussian function, or normal distribution 18.2 Level curves Exercises Math drills for practice Scientific applications Chapter 19 Partial derivatives 19.1 Slopes of a surface 19.2 Partial differentiation 19.3 The chain rule for multiple variables 19.4 Partial derivatives and uncertainty analysis 19.5 Higher order partial derivatives Exercises Math drills for practice Scientific applications Chapter 20 Extrema of functions of two (or more) variables 20.1 Maximum and minimum points 20.2 Saddle points Exercises Math drills for practice Scientific applications Part VI Integration Chapter 21 The area under a curve 21.1 Geometric intuition 21.2 Notation: the integral sign 21.3 Riemann sums 21.4 The Fundamental Theorems of Calculus Exercises Math drills for practice Scientific applications Chapter 22 Calculating antiderivatives and areas 22.1 Antiderivatives are not unique 22.2 Indefinite integrals 22.3 Basic formulas 22.3.1 Integrating a constant multiple of a function 22.3.2 Integrating sums and differences 22.4 Calculating areas underneath graphs 22.5 Integrating vectors 22.6 Velocity and distance 22.7 The average value of a function 22.7.1 Geometric interpretation of the average value 22.8 Work 22.8.1 Non-constant force 22.8.2 PV work Exercises Math drills for practice Scientific applications Chapter 23 Integration techniques 23.1 Integration by substitution 23.2 Integration by parts 23.3 The LATE rule 23.4 Looking it up or using a computer Exercises Chapter 24 Numerical integration 24.1 Some pretend data 24.2 The trapezoid method 24.3 The special case of equal intervals Exercises Scientific applications Part VII Differential Equations Chapter 25 First-order ordinary differential equations 25.1 Differential equations from the realworld 25.2 Initial conditions 25.3 Separation of variables 25.3.1 The general form of a separable equation 25.3.2 A shortcut 25.4 Using a computer 25.5 Qualitative analysis Exercises Math drills for practice Scientific applications Chapter 26 Numerical solutions of differential equations 26.1 Euler’s method 26.2 Using computer packages Exercises Math drills for practice Scientific applications Part VIII Probability Chapter 27 Probability foundations 27.1 Sample space 27.2 Events 27.2.1 Events in pictures 27.3 Probability 27.3.1 Properties of a probability distribution 27.3.2 Probability notation 27.4 Probability distributions in pictures 27.5 Tables of counts 27.5.1 Subsetting the table of counts Exercises Math drills for practice Scientific applications Chapter 28 Random variables 28.1 Standard notation for random variables 28.1.1 Useful rules 28.2 Discrete and continuous 28.3 The probability function 28.4 Cumulative distribution function 28.4.1 Interval probabilities 28.4.2 Warning: endpoints Exercises Math drills for practice Scientific applications Chapter 29 Binomial distribution 29.1 Bernoulli trials 29.1.1 Experiments with two outcomes 29.1.2 Independence Probabilities for independent events 29.1.3 Sequence of Bernoulli trials 29.2 Binomial distribution 29.3 Shape of the binomial distribution 29.4 Binomial probability function 29.4.1 Number of combinations 29.4.2 Formula for the binomial probability function Some important things to note 29.5 Binomial probabilities by computer Exercises Math drills for practice Scientific applications Chapter 30 Conditional probability 30.1 Conditional probability: shrinking the sample space 30.2 Conditional probability in pictures 30.3 Conditionals and intersections 30.3.1 Everyday language for conditionals and intersections 30.3.2 The Multiplication Rule 30.4 Bayes’ theorem for inverting conditionals 30.5 Statistical independence 30.5.1 Independence of everyday events 30.5.2 Independence in pictures 30.5.3 Independence for random variables Exercises Math drills for practice Scientific applications Chapter 31 The total probability rule 31.1 Total probability of an event 31.1.1 The partition theorem Example 31.1.2 Warning: a common mistake 31.2 The partition theorem in pictures 31.3 Some examples Worked Example Solution Worked Example Solution Worked Example Solution Exercises Math drills for practice Scientific applications Part IX Statistical Inference Chapter 32 Hypothesis tests 32.1 What is statistical inference? 32.1.1 Populations and samples 32.1.2 But what does a statistical inferrer actually do? 32.2 The world’s simplest hypothesis test 32.2.1 Can I make a coin flip go my way? 32.2.2 How lucky is 9 heads out of 10 flips? 32.2.3 Quantifying luckiness: the P-value calculation 32.2.4 What can you conclude? Interpreting the P-value 32.2.5 P-values as tail probabilities 32.2.6 Why do we need to use tail probabilities? 32.2.7 Is it significant? 32.3 Principles of hypothesis testing 32.3.1 Null hypothesis and alternative hypothesis 32.3.2 P-values 32.3.3 A spectrum of P-values, or an accept/reject directive? 32.3.4 Statistical significance 32.3.5 Significance level as a false positive rate 32.3.6 Two-sided or one-sided tests? Exercises Math drills for practice Scientific applications Chapter 33 Presidents, deep-sea divers, and sports stars 33.1 Presidential hypothesis test Are presidents more likely to have sons? 33.2 Deep-sea divers hypothesis test Why should deep-sea divers have daughters? 33.3 Sports stars hypothesis test Presenting your hypothesis test Hypothesis test for sports stars Why do sports stars need the right birthday? 33.4 The role of hypothesis testing in science Exercises Math drills for practice Scientific applications Chapter 34 Estimation and likelihood 34.1 Estimation 34.2 Likelihood 34.2.1 What’s the likelihood of X=125 under different values of p? 34.2.2 Formula for the likelihood function 34.2.3 What happens if we change x? 34.2.4 Difference between the likelihood and the probability function 34.3 Finding the maximum likelihood estimate 34.3.1 The hat notation for an estimate 34.3.2 Summary of the maximum likelihood procedure 34.4 Estimators 34.4.1 Maximum likelihood estimator for Binomial(n, p) 34.5 The role of likelihood in scientific modelling Exercises Math drills for practice Scientific applications Part X Discrete Probability Distributions Chapter 35 Simulation and visualisation 35.1 Simulation 35.1.1 Random numbers in R 35.2 Histograms 35.2.1 Effect of sample size 35.3 Histograms as empirical probability functions 35.3.1 Standardising the histogram to the probability scale Exercises Math drills for practice Scientific applications Chapter 36 Mean 36.1 The distribution mean 36.1.1 The distribution mean as the average of many observations 36.1.2 A formula for the distribution mean 36.1.3 Multiple notations for the mean 36.2 Binomial distribution mean 36.3 Combining random variables with constants 36.3.1 Adding and multiplying by constants 36.3.2 Expectation of transformed random variables 36.3.3 Simulating transformed random variables 36.4 Combining random variables 36.5 Expectation of X2 and other transformations 36.6 Binomial mean explained 36.7 Mean of estimators Exercises Math drills for practice Scientific applications Chapter 37 Variance 37.1 The distribution variance 37.1.1 An alternative formula for variance 37.2 Variance properties 37.2.1 Adding and multiplying by constants 37.2.2 Adding random variables together 37.3 Binomial distribution variance 37.4 Standard deviation 37.5 Variance of estimators 37.5.1 Standard error 37.5.2 Margin of error and confidence intervals Exercises Math drills for practice Scientific applications Chapter 38 Discrete probability models 38.1 Binomial distribution 38.2 Poisson distribution 38.2.1 Poisson distribution in concept 38.2.2 Modelling with the Poisson distribution 38.2.3 Mean and variance 38.2.4 Shape 38.2.5 R commands 38.2.6 Sum of Poisson random variables 38.3 Inference with the Poisson distribution 38.3.1 Maximum likelihood estimation with the Poisson distribution 38.3.2 Standard error and confidence intervals 38.3.3 Estimation from many independent observations Method 1: multiplying likelihoods Method 2: sum of Poissons is Poisson 38.3.4 Hypothesis testing with the Poisson distribution 38.4 Geometric distribution 38.4.1 Mean and variance 38.4.2 Shape 38.4.3 R commands 38.4.4 Sum of Geometric random variables 38.4.5 Likelihood 38.4.6 Hypothesis testing with the geometric distribution 38.5 Negative binomial distribution 38.5.1 Mean and variance 38.5.2 Shape 38.5.3 R commands 38.5.4 Sum of negative binomial random variables 38.5.5 Hypothesis testing with the negative binomial distribution 38.5.6 Likelihood and estimation Exercises Math drills for practice Scientific applications Part XI Continuous Probability Distributions Chapter 39 Continuous random variables 39.1 What does it mean to be continuous? 39.2 Probability density function (PDF) 39.2.1 Explicit definition of the PDF 39.3 Calculating probabilities 39.3.1 Finding the CDF from the PDF 39.4 Inference with continuous random variables 39.5 Mean and variance 39.5.1 Expectation of X2 and other transformations 39.5.2 Variance 39.5.3 Properties of expectation and variance Exercises Math drills for practice Scientific applications Chapter 40 Common continuous probability models 40.1 Uniform distribution 40.1.1 PDF and CDF 40.1.2 R commands 40.1.3 Mean and variance 40.1.4 Inference with the uniform distribution 40.2 Exponential distribution 40.2.1 Poisson process 40.2.2 Waiting time in the Poisson process 40.2.3 Mean and variance 40.2.4 Memorylessness 40.2.5 Summary of the Poisson process 40.2.6 R commands for the exponential distribution 40.2.7 Sum of exponential random variables 40.2.8 Likelihood and hypothesis testing 40.3 Gamma distribution 40.3.1 Mean and variance 40.3.2 PDF and CDF 40.3.3 R commands 40.3.4 Sum of gamma random variables 40.3.5 Likelihood and estimation 40.4 Inference with the exponential distribution 40.4.1 Maximum likelihood estimation Method 1: multiplying likelihoods Method 2: sum of exponentials is gamma 40.4.2 Hypothesis testing Exercises Math drills for practice Scientific applications Chapter 41 Normal distribution and inference 41.1 Normal distribution 41.1.1 Mean and variance 41.1.2 PDF and CDF 41.1.3 R commands 41.1.4 Adding and multiplying by constants 41.1.5 Probability intervals 41.1.6 Sum of normal random variables 41.1.7 Likelihood and hypothesis testing 41.2 The central limit effect 41.2.1 Normality of the sample mean, 41.2.2 Why does the central limit effect work? 41.3 One-size-fits-all statistical inference 41.3.1 Hypothesis test for the distribution mean 41.3.2 Confidence intervals for the distribution mean 41.3.3 t-tests and confidence intervals in R 41.3.4 Inference on other parameters Exercises Math drills for practice Scientific applications Part XII Linear Regression Chapter 42 Fitting linear functions: theory and practice 42.1 Finding relationships between variables 42.2 Key questions 42.3 The simple linear model 42.3.1 The explained part of the model 42.3.2 The scatter model 42.3.3 Best-fit line 42.4 The method of least squares 42.4.1 Why do we use the least-squares criterion? 42.5 Fitting a linear function to data; a simple example 42.6 Fitting a linear function to data; a more complex example 42.7 Fitting the simple linear model using R 42.7.1 Creating a pretend data set 42.7.2 Using a larger sample size 42.7.3 Checking assumptions using residual plots Exercises Math drills for practice Scientific applications Chapter 43 Quantifying relationships 43.1 Finding P-values using R 43.2 False positives, or Type I errors 43.3 False negatives, or Type II errors 43.4 Confidence intervals 43.4.1 Interpreting the confidence interval Exercises References Index