Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials
Definition
For , a Bernoulli random variable takes the value with probability and with probability .
For , a binomial random variable is a sum of mutually independent Bernoulli variables. When , this is the constant zero random variable.
Depends on
Used by
- Critical-reaction binomial guarded tau-leaping Algorithm
- De Moivre-Laplace central limit theorem Corollary
- The claimed univariate clean-quantile count exponent is too strong Counterexample
- The Erdős-Rényi finite random graph G(n,p) Definition
- A biased estimator can have lower MSE Example
- A binomial guard prevents overdraw but changes Poisson noise Example
- A holdout estimate targets a smaller training sample Example
- A Lindeberg array with no identically distributed row Example
- A one-observation Bernoulli-logit Fisher-scoring step Example
- Bernoulli sample frequencies Example
- Complete separation prevents a finite logistic MLE Example
- Equal predictive risks do not identify a model label Example
- First- and second-moment bounds for a nonempty Bernoulli random subset Example
- Normal approximation to binomial probabilities Example
- Paired K-fold comparison retains correlated fold errors Example
- Random row splits leak repeated-group information Example
- Rank deficiency gives singular information and nonunique weighted least squares Example
- Rao–Blackwellizing one Bernoulli observation by the sample sum Example
- The Bernoulli identity-link boundary leaves the regular family Example
- The binomial sample proportion Example
- Weakly close root laws can still give failed uniform coverage Example
- A Bernoulli(p) variable has mean p and variance p(1-p); a binomial(n,p) variable has mean np and variance np(1-p) Lemma
- Bernoulli root laws are uniformly close in Levy–Prokhorov distance Lemma
- Chernoff bound for independent bernoulli trials Lemma
- One-sided concentration bound for independent median trials Lemma
- The Binomial count has its usual probability mass function Lemma
- m-out-of-n bootstrap and subsampling recover the uniform endpoint root Theorem
- No universal unbiased estimator of K-fold cross-validation variance Theorem
- Univariate trimmed-mean bound under sample replacement Theorem
Dependency tree · two levels
19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 5.1 (standard reference, not scraped)
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 3.1.5 (standard reference, not scraped)