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Event independence
Two events are independent when their intersection probability equals the product of their marginal probabilities; this definition does not rely on conditioning on a zero-probability event.
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
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Graph walks and simple paths
In a finite undirected simple graph, a walk requires each consecutive pair of vertices to be adjacent; a simple path additionally requires all vertices to be distinct.
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Finite commutative ring
A finite unital ring is commutative when multiplication commutes for every pair of elements; zero divisors may still exist.
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Finite two-sided ideal
A subset of a finite ring is a two-sided ideal when it is an additive subgroup and absorbs multiplication by parent-ring elements on both sides.
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Real Banach space
A real normed vector space is a Banach space when it is complete in the metric induced by its own norm.
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Real inner-product space
A symmetric, positive-definite real bilinear form is a real inner product; positive semidefiniteness or positive sampled values do not suffice.
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Bounded linear operator
A linear map between normed spaces is bounded when one nonnegative constant uniformly controls all output norms relative to input norms.
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Independence of random variables
Under one finite joint law, independence requires every value-pair mass to factor as the product of its marginal masses.
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Connected component
In a finite undirected simple graph, connected components are exactly the vertex equivalence classes defined by the existence of a path.
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Finite integral domain
A finite commutative unital ring with 1 distinct from 0 is an integral domain when it has no nonzero zero divisors.
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Polynomial ring over a finite commutative ring
Ordinary polynomials over a finite commutative coefficient ring have finite-support coefficient sequences, but the collection over all finite degrees is still infinite.
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Inner-product-induced norm
On the same real inner-product space, the norm is the nonnegative square root of the self-inner-product; the triangle inequality uses the universal Cauchy–Schwarz certificate.
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Finite conditional expectation
Relative to a partition-generated sigma-algebra on a finite probability space, conditional expectation takes the weighted mean on each positive-mass block and preserves block integrals.
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Finite filtration
A finite-time filtration is a sequence of sigma-algebras on the same probability space whose information increases with time.
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Independent identically distributed sequence
An IID sequence requires every variable to have the same distribution and every finite subfamily to be mutually independent.
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Equivalence class
For a fixed complete equivalence relation, the class of an element a consists of every carrier element related to a.
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Finite field
A finite commutative unital ring is a finite field when every nonzero element has a multiplicative inverse.
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Real Hilbert space
A real inner-product space is Hilbert when it is complete in the norm and metric induced by that same inner product.
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Finite martingale
An integrable finite process adapted to the same filtration is a martingale when the next value's conditional expectation equals the current value almost surely at every step.
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Quotient set
The distinct equivalence classes of a complete equivalence relation form its quotient set and partition the original carrier.
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Cardinality comparison of sets
An injection, surjection or bijection between explicit finite source and target sets supplies a corresponding finite cardinality-comparison certificate.
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Cumulative distribution function
For a real-valued random variable, F(t)=P(X≤t) is nondecreasing and right-continuous; in a finite discrete law, its jump is the point mass.
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Convergence in probability
X_n converges in probability to X when, for every epsilon>0, the probability of deviation greater than epsilon tends to zero.
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At-most-countable set
A set is at most countable if it is finite or has an explicit injection into the natural numbers; this is not the same as countably infinite.