Probability theory and stochastic processes Definition
Independent and identically distributed random variables
Builds on Independence of random variablesIdentically distributed random variables
Definition
Let be a family of random variables on one probability space, all with values in the same measurable space . The family is independent and identically distributed (i.i.d.) if both of these conditions hold:
- The family is independent.
- The random variables are identically distributed: all have the same law .
Remarks
One also says that is an i.i.d. sequence with common law , or a random sample from . The results of repeated tosses of one coin form an i.i.d. sequence of Bernoulli random variables. The law of large numbers and the central limit theorem are first proved for i.i.d. sequences.
Reading path
To understand this page from the ground up, read these 47 pages in order. The foundations come first.
- Construction of the reals from Cauchy sequences concept
- Element definition
- Integers object
- Ordered field definition
- Proposition concept
- Random experiment concept
- Rational numbers object
- Real numbers object
- Set definition
- Ring of integers statement
- Disjunction definition
- Implication definition
- Predicate definition
- Ordered pair definition
- Union definition
- Sample space definition
- The rationals form a field statement
- Existential quantifier definition
- Universal quantifier definition
- Cartesian product definition
- Subset definition
- The rationals form an ordered field statement
- Uniqueness quantifier definition
- Binary relation definition
- Universal set definition
- Order on the reals definition
- Power set definition
- Complement definition
- Function definition
- Interval definition
- Preimage of a set definition
- Sigma-algebra definition
- Event definition
- Borel sigma-algebra definition
- Probability axioms concept
- Probability measure definition
- Probability distribution definition
- Probability space definition
- Conditional probability definition
- Random variable definition
- Independence of two events definition
- Law of a random variable definition
- Mutual independence of events definition
- Identically distributed random variables definition
- Independent sigma-algebras definition
- Independence of random variables definition
- Independent and identically distributed random variables definition
Where this leads
131 pages build on this one.
- Bias of an estimator
- Bernoulli process
- Statistic
- Likelihood function
- De Moivre–Laplace theorem
- Statistical hypothesis test
- Method of moments
- Weak law of large numbers
- Random walk
- Strong law of large numbers
- Neyman–Pearson lemma
- P-value
- Chi-squared test
- Central limit theorem
- Maximum likelihood estimation
- Galton–Watson process
- Student's t-test
- Sample variance
- Bayesian inference
- Confidence interval
- Student's t-distribution
- Sample mean
- Type I and type II errors
- Chi-squared distribution
- Estimator
- Mean squared error
- Gambler's ruin
- Consistent estimator
- Law of large numbers
- Bernstein–von Mises theorem
- Efficiency of an estimator
- Log-rank test
- Poisson regression
- Sequential analysis
- Ergodic decomposition
- Koksma–Hlawka inequality
- Hodges' superefficient estimator
- Mathematical statistics
- F-test
- Wald–Wolfowitz optimality of the sequential probability ratio test
- Sequential probability ratio test
- Consistency of the maximum likelihood estimator
- Bayes factor
- Pickands–Balkema–De Haan theorem
- Basu's theorem
- Order statistic
- Lindley's paradox
- M-estimator
- Kernel regression
- Score test
- Kingman's subadditive ergodic theorem
- Multivariate analysis of variance
- Asymptotic normality of the maximum likelihood estimator
- Mann–Whitney U test
- Score
- F-distribution
- Edgeworth expansion
- Monotone likelihood ratio
- Sample mean and sample covariance
- Wilcoxon signed-rank test
- Minimal sufficient statistic
- Whittle likelihood
- Logistic regression
- Cramér–von Mises criterion
- Kernel density estimation
- Erdős–Kac theorem
- Robust statistics
- Bias–variance decomposition
- Monte Carlo integration
- Breakdown point
- Delta method
- Noncentral chi-squared distribution
- Hájek–Le Cam convolution theorem
- Behrens–Fisher problem
- Hodges–Lehmann estimator
- Kruskal–Wallis test
- Sampling distribution
- Wishart distribution
- Statistical inference
- Minimum-variance unbiased estimator
- Sign test
- Empirical likelihood
- Dvoretzky–Kiefer–Wolfowitz inequality
- Bayesian information criterion
- Berry–Esseen theorem
- Cochran's theorem
- Jackknife
- Birkhoff's ergodic theorem
- Akaike information criterion
- Credible interval
- Independence of sample mean and sample variance for normal samples
- Estimating equations
- Anderson–Darling test
- Family-wise error rate
- Dickey–Fuller test
- Generalized method of moments
- Uniformly most powerful test
- Empirical distribution function
- Bootstrap
- Oseledets theorem
- Ancillary statistic
- Complete statistic
- Horvitz–Thompson estimator
- U-statistic
- L-estimator
- Power of a test
- Sufficient statistic
- Missing data mechanisms
- Likelihood-ratio test
- Karlin–Rubin theorem
- Conjugate prior
- Kolmogorov–Smirnov test
- Expectation–maximization algorithm
- Hotelling's T-squared distribution
- Cross-validation
- Wald test
- Generalized linear model
- Box–Jenkins method
- Pesin theory
- Binomial proportion confidence interval
- False discovery rate and the Benjamini–Hochberg procedure
- Neyman construction
- Wilks' theorem
- Permutation test
- Fisher's exact test
- Bonferroni correction
- Frequentist inference
- Likelihood principle
- Fisher–Neyman factorization theorem
- Observed information
- Point estimation
Further reading
- Web https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables, Definition for more than two random variables
About this page
| Tag | 0AJN7WHBJZ |
|---|---|
| URI | m2://0AJN7WHBJZ |
| Kind | definition |
| Writing | W0 stub |
| MSC | 60A05 |
| Defines | Independent and identically distributed random variables |
| Uses | Independence of random variables, Identically distributed random variables |
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What links here
Used by
- Bernoulli process
- Statistic
- Likelihood function
- Weak law of large numbers
- Random walk
- Strong law of large numbers
- Central limit theorem
- Galton–Watson process
- Sample mean
- Chi-squared distribution
- Estimator
- Law of large numbers
- Bernstein–von Mises theorem
- Hodges' superefficient estimator
- Consistency of the maximum likelihood estimator
- Pickands–Balkema–De Haan theorem
- Kernel regression
- Asymptotic normality of the maximum likelihood estimator
- Mann–Whitney U test
- Edgeworth expansion
- Wilcoxon signed-rank test
- Cramér–von Mises criterion
- Kernel density estimation
- Robust statistics
- Kruskal–Wallis test
- Wishart distribution
- Sign test
- Empirical likelihood
- Dvoretzky–Kiefer–Wolfowitz inequality
- Independence of sample mean and sample variance for normal samples
- Estimating equations
- Anderson–Darling test
- Generalized method of moments
- Empirical distribution function
- Bootstrap
- U-statistic
- Kolmogorov–Smirnov test
- Wilks' theorem