دانلود کتاب Mathematics and Statistics for Science

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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