Probability theory and stochastic processes Definition
Identically distributed random variables
Builds on Law of a random variable
Definition
Let be a set with a σ-algebra. Let be a random variable on a probability space with values in . Let be a random variable on a probability space with values in . and are identically distributed if they have the same law: for each . One writes .
Remarks
For real random variables, this holds if and only if and have the same cumulative distribution function. Identically distributed random variables need not be equal. For a fair coin, the indicator of heads and the indicator of tails are identically distributed, but they are never equal. If and is a measurable function, then .
Reading path
To understand this page from the ground up, read these 41 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
- Random variable definition
- Law of a random variable definition
- Identically distributed random variables definition
Where this leads
132 pages build on this one.
- Bias of an estimator
- Bernoulli process
- Statistic
- Likelihood function
- De Moivre–Laplace theorem
- Statistical hypothesis test
- Method of moments
- Independent and identically distributed random variables
- 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 two random variables
About this page
| Tag | 01X8YSDTBW |
|---|---|
| URI | m2://01X8YSDTBW |
| Kind | definition |
| Writing | W0 stub |
| MSC | 60E05 |
| Defines | Identically distributed random variables |
| Uses | Law of a random variable |
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