Integers
Description
The integers are the numbers . We write for the set of integers. The set contains the natural numbers and their negatives . With addition and multiplication, is a commutative ring. Each integer has an additive inverse , thus subtraction is always possible in . The integers have a total order that extends the order of .
Remarks
The letter comes from the German word "Zahlen", which means "numbers". One construction of the integers uses pairs of natural numbers. The rational numbers contain the integers as the fractions .
Where this leads
451 pages build on this one.
- Variance
- Prior probability
- Absolute value
- exp z = 1 iff z is an integer multiple of 2πi
- Chernoff bound
- Chapman–Kolmogorov equation
- Bias of an estimator
- Identically distributed random variables
- Law of the unconscious statistician
- Density of the normal distribution
- Exponential distribution
- Covariance matrix
- Chebyshev's inequality
- Characteristic function of a sum of independent random variables
- Stationary distribution
- Bernoulli process
- Order on the reals
- Markov chain
- Statistic
- Law of total variance
- Variance of the normal distribution
- Triangle inequality
- Cauchy–Schwarz inequality for expectations
- Monotonicity of probability
- Every real number has a decimal expansion
- Correlation coefficient
- Variance under translation
- Density of the irrationals
- Almost sure convergence
- Dedekind cut
- Law of total expectation
- Probability density function
- Expectation of a product of independent random variables
- Likelihood function
- Markov's inequality
- Standard score
- Poisson point process
- Gamma distribution
- De Moivre–Laplace theorem
- Buffon's needle problem
- Kolmogorov's zero–one law
- Independent random variables are uncorrelated
- Convolution of probability distributions
- Normal distribution
- Mean of the Poisson distribution
- Statistical hypothesis test
- Independence means the joint law is the product of the marginals
- Median
- Geometric probability
- Intermediate value theorem for connected spaces
- Dirac measure
- Method of moments
- Mean of the normal distribution
- The rationals form an ordered field
- Independent sigma-algebras
- The reals are Archimedean
- Independent and identically distributed random variables
- Classical definition of probability
- Pairwise independent events that are not mutually independent
- Ceiling function
- Marginal distribution
- Moment-generating function of the normal distribution
- Quantile function
- Connected subsets of the real line are intervals
- Weak law of large numbers
- Multiplication rule
- Borel sigma-algebra
- Ring of integers
- Addition rule for two events
- Random walk
- Random variable with a density
- Uncorrelated random variables that are not independent
- Reverse triangle inequality
- Kurtosis
- Martingale
- Variance of a sum
- Mean of the geometric distribution
- Expected value
- Strong law of large numbers
- Covariance
- Random variable
- Base rate fallacy
- Law of total probability
- Independence of two random variables
- Neyman–Pearson lemma
- Convergence in probability
- Boole's inequality
- P-value
- Chi-squared test
- Convergence in probability implies convergence in distribution
- Central limit theorem
- 68–95–99.7 rule
- Transition matrix
- Conditional variance
- Order on the integers
- Conditional probability
- Lévy's continuity theorem
- Convergence of random variables
- Maximum likelihood estimation
- Multivariate normal distribution
- Mutual independence of events
- Absolute value is multiplicative
- Skewness
- Galton–Watson process
- Markov property
- Induction on the integers
- Student's t-test
- Discrete random variable
- Bernoulli distribution
- Convergence in distribution
- The cumulative distribution function determines the distribution
- Sample variance
- Bayesian inference
- Confidence interval
- Posterior probability
- Variance under scaling
- Discrete uniform distribution
- Coupon collector's problem
- Cumulative distribution function
- Characteristic function
- The rationals form a field
- Pairwise independence
- Cauchy distribution
- Bayes' theorem
- Variance as the mean of the square minus the square of the mean
- Birthday problem
- Moment-generating function
- Student's t-distribution
- Characteristic function of the normal distribution
- Second Borel–Cantelli lemma
- Sum of independent normal random variables
- Independence of two events
- Linearity of expectation
- Monty Hall problem
- Moment
- Beta distribution
- Probability of the complement
- Expectation with respect to the binomial distribution
- Cumulant generating function
- Joint probability distribution
- Probability distribution
- Stochastic process
- Sample mean
- The reals form an ordered field
- Type I and type II errors
- Almost surely
- Fraction in lowest terms
- Characterization of the floor
- Generating function of a sum of independent random variables
- Chi-squared distribution
- Limit of the cumulative distribution function at infinity
- Hypergeometric distribution
- Moments are derivatives of the moment-generating function at zero
- Density of the rationals
- Almost sure convergence implies convergence in probability
- Bertrand paradox
- First Borel–Cantelli lemma
- The characteristic function determines the distribution
- Interval
- Law of a random variable
- Convergence in mean
- Estimator
- Variance of a sum of independent random variables
- Mean squared error
- Poisson distribution
- Probability measure
- Binomial distribution
- Standard deviation
- Law of total probability for an event and its complement
- Probability mass function
- Gambler's ruin
- Continuous uniform distribution
- There is a natural number n with 1/n below any positive epsilon
- Hoeffding's inequality
- Moment-generating function of a sum of independent random variables
- Continuity of probability measures
- Independence of random variables
- Geometric distribution
- Monotonicity of the cumulative distribution function
- Consistent estimator
- Poisson limit theorem
- Negative binomial distribution
- A root of an integer is an integer or irrational
- Tail sum formula for expectation
- Conditional expectation
- Absorbing Markov chain
- Sum of independent Poisson random variables
- Conditional probability distribution
- Floor function
- Probability-generating function
- Memorylessness
- Conditional independence
- Path-connected spaces are connected
- Central moment
- Law of large numbers
- Probability space
- Doob's upcrossing inequality
- Bernstein–von Mises theorem
- Cramér–Rao bound
- Efficiency of an estimator
- Log-rank test
- Functional data analysis
- Poisson regression
- Sequential analysis
- Ergodic decomposition
- Gauss circle problem
- Koksma–Hlawka inequality
- Hodges' superefficient estimator
- Martingale representation theorem
- Absorbing set
- Mathematical statistics
- F-test
- Laplace transform of the Dirac delta
- Stationary process
- A translation of the integers is not a homomorphism
- Orthogonal polynomials on the unit circle
- Cramér–Wold theorem
- Boolean model
- Stochastic partial differential equation
- Wald–Wolfowitz optimality of the sequential probability ratio test
- Doob's martingale inequality
- Seminorm
- Sequential probability ratio test
- The integers modulo the even integers
- Brownian bridge
- Consistency of the maximum likelihood estimator
- Wiener process
- Empirical Bayes method
- Karhunen–Loève theorem
- Brent's method
- Itô isometry
- Absolute continuity
- Bayes factor
- Pickands–Balkema–De Haan theorem
- Basu's theorem
- Fundamental group of the circle
- Variogram
- Robbins–Monro algorithm
- Contiguity of measures
- Order statistic
- Balanced set
- Dirichlet process
- Lindley's paradox
- Universally measurable set
- M-estimator
- Ornstein–Uhlenbeck process
- Kernel regression
- Bayes estimator
- Bisection method
- Waring's problem
- Adapted process
- Ramanujan tau function
- Score test
- Kingman's subadditive ergodic theorem
- Multivariate analysis of variance
- McDiarmid's inequality
- Lévy's characterization of Brownian motion
- Asymptotic normality of the maximum likelihood estimator
- Mann–Whitney U test
- Kolmogorov automorphism
- The even integers form a subgroup of the integers
- Score
- Arbitrary-precision arithmetic
- F-distribution
- Edgeworth expansion
- Crofton formula
- Monotone likelihood ratio
- Sample mean and sample covariance
- Wilcoxon signed-rank test
- Metropolis–Hastings algorithm
- Subderivative
- Continuous-time Markov chain
- Darboux function
- Kriging
- Log-convexity of the gamma function
- Identifiability
- Baire function
- Minimal sufficient statistic
- Non-surjective epimorphism of algebras
- Whittle likelihood
- Logistic regression
- Convergence to equilibrium of ergodic Markov chains
- Presburger arithmetic
- Cramér–von Mises criterion
- Azuma's inequality
- Gaussian free field
- Parry measure
- Kernel density estimation
- The cosets of 3Z in the integers
- Reduction modulo n is a homomorphism
- Erdős–Kac theorem
- Detailed balance
- Coupling from the past
- Robust statistics
- Hölder's theorem on Archimedean groups
- Hidden Markov model
- Finite-dimensional distributions
- Continuous mapping theorem
- Jacobi's four-square theorem
- Bias–variance decomposition
- Lévy process
- Wald's equation
- Random ordinary differential equation
- Monte Carlo integration
- Time series
- Optional stopping theorem
- Measurable space
- Breakdown point
- Delta method
- Minimum description length
- Noncentral chi-squared distribution
- Hájek–Le Cam convolution theorem
- Logarithmically convex function
- Harmonic measure
- Behrens–Fisher problem
- Submartingale
- Hodges–Lehmann estimator
- Kruskal–Wallis test
- Stochastic difference equation
- Sampling distribution
- Information geometry
- Recurrence and transience of states
- Malliavin calculus
- Wishart distribution
- Birth–death process
- Karamata's inequality
- Bayes risk
- Statistical inference
- Minimum-variance unbiased estimator
- Sign test
- Lattice-ordered group
- Slutsky's theorem
- Tanaka's formula
- Geometric Brownian motion
- Empirical likelihood
- Dvoretzky–Kiefer–Wolfowitz inequality
- Bayesian information criterion
- Berry–Esseen theorem
- Continuity of measure from below and above
- Legendre transformation
- Hunt process
- Cochran's theorem
- Stochastic approximation
- Jackknife
- Quantile regression
- Markov chain mixing time
- Birkhoff's ergodic theorem
- Akaike information criterion
- Hewitt–Savage zero–one law
- Jensen's inequality for integrals
- Statistical model
- Bramble–Hilbert lemma
- Reflection principle
- Credible interval
- Conformal modulus
- Independence of sample mean and sample variance for normal samples
- Self-similar process
- Gibbs sampling
- Estimating equations
- Supermartingale
- Anderson–Darling test
- Spearman's rank correlation coefficient
- Family-wise error rate
- Torsion group
- Markov chain Monte Carlo
- Nested intervals theorem
- Dickey–Fuller test
- Generalized method of moments
- Doob's martingale convergence theorems
- Random integral equation
- Uniformly most powerful test
- Bernstein inequalities
- Kendall rank correlation coefficient
- Semimartingale
- Empirical distribution function
- Bootstrap
- Rounding
- Minimax approximation
- Influence function
- Oseledets theorem
- Itô calculus
- Ancillary statistic
- Stopped process
- Jeffreys prior
- Moments of the log-normal distribution
- Complete statistic
- Horvitz–Thompson estimator
- Binary quadratic form
- U-statistic
- Fractional Brownian motion
- Bayesian hierarchical model
- L-estimator
- Doubling on the integers is not an isomorphism
- Power of a test
- Parametric model
- Left-orderable group
- Randomization
- Sufficient statistic
- Gaussian process
- Missing data mechanisms
- Likelihood-ratio test
- Floating-point arithmetic
- Karlin–Rubin theorem
- Hardy–Littlewood circle method
- Conjugate prior
- Kolmogorov–Smirnov test
- Expectation–maximization algorithm
- Hotelling's T-squared distribution
- Burkholder–Davis–Gundy inequality
- Doob–Meyer decomposition theorem
- Cross-validation
- Barbier's theorem
- Wald test
- Integral quadratic form
- Extremal length
- Pivotal quantity
- Generalized linear model
- Random graph
- Tame Fréchet space
- Box–Jenkins method
- Kolmogorov continuity theorem
- The odd integers are not a subgroup of the integers
- Shrinkage estimator
- Pesin theory
- Torsion-free group
- Empirical process
- Binomial proportion confidence interval
- Lévy's upward theorem
- False discovery rate and the Benjamini–Hochberg procedure
- Particle filter
- Neyman construction
- Wilks' theorem
- Differentiable curve
- Skorokhod's embedding theorem
- Blaschke–Lebesgue theorem
- Permutation test
- Fisher's exact test
- Bonferroni correction
- Doob decomposition theorem
- Local asymptotic normality
- Frequentist inference
- Likelihood principle
- Chebyshev nodes
- Fisher–Neyman factorization theorem
- Girsanov theorem
- Quadratic variation
- Observed information
- Feller process
- Point estimation
- Local martingale
- Orderable group
Further reading
- Web https://en.wikipedia.org/wiki/Integer, lead section
- Web https://en.wikipedia.org/wiki/Integer, Algebraic properties
About this page
| Tag | 0C4JS9RF5M |
|---|---|
| URI | m2://0C4JS9RF5M |
| Aliases | integer, set-of-integers |
| Kind | object |
| Writing | W1 stated |
| MSC | 11A99 |
| Same as |
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What this page needs for W2
- section Invariants: a table of the main invariants (0 of 1)
- edge instance_of out (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Used by
- exp z = 1 iff z is an integer multiple of 2πi
- Ceiling function
- Ring of integers
- Order on the integers
- Induction on the integers
- Fraction in lowest terms
- A root of an integer is an integer or irrational
- Floor function
- Gauss circle problem
- A translation of the integers is not a homomorphism
- The integers modulo the even integers
- Fundamental group of the circle
- Waring's problem
- Ramanujan tau function
- The even integers form a subgroup of the integers
- Arbitrary-precision arithmetic
- Non-surjective epimorphism of algebras
- Presburger arithmetic
- The cosets of 3Z in the integers
- Reduction modulo n is a homomorphism
- Jacobi's four-square theorem
- Torsion group
- Rounding
- Binary quadratic form
- Doubling on the integers is not an isomorphism
- Floating-point arithmetic
- Integral quadratic form
- The odd integers are not a subgroup of the integers
- Torsion-free group