User talk:Mindey/MathNotes

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Newton Binomial[edit]

Notation of Combinations[edit]

A property of Combinations:[edit]

Integral of 1/x[edit]

Normal law density and CDF[edit]

PDF:

CDF:

, where

Continuous r.v. versus Absolutely continuous r.v.[edit]

is continuous r.v.

is absolutely continous r.v. , or, in discrete case:

Poisson integral[edit]

Integration by parts heuristic[edit]

If u = u(x), v = v(x), and the differentials du = u '(xdx and dv = v'(xdx, then integration by parts states that

Liate rule

A rule of thumb proposed by Herbert Kasube of Bradley University advises that whichever function comes first in the following list should be u:[1]

L - Logarithmic functions: ln x, logb x, etc.
I - Inverse trigonometric functions: arctan x, arcsec x, etc.
A - Algebraic functions: x2, 3x50, etc.
T - Trigonometric functions: sin x, tan x, etc.
E - Exponential functions: ex, 19x, etc.

The function which is to be dv is whichever comes last in the list: functions lower on the list have easier antiderivatives than the functions above them. The rule is sometimes written as "DETAIL" where D stands for dv.

Probability of difference of events[edit]

Definition of Measurable Function = Measurable Mapping ?[edit]

Let and be measurable spaces, meaning that and are sets equipped with respective sigma algebras and . A function

is said to be measurable if for every . The notion of measurability depends on the sigma algebras and . To emphasize this dependency, if is a measurable function, we will write

— Preceding unsigned comment added by 128.211.164.79 (talk) 02:13, 22 August 2012 (UTC)

Lp space[edit]

From undergrad notes: space, where is a space of sequences, where the distance between the sequences is computed with formula . The space will constitute of the sequences with the property . In other words, this space will be made of sequences, such that their distance from the zero sequence is finite.

From Wikipedia: a function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. Let 1 ≤ p < ∞ and (S, Σ, μ) be a measure space. Consider the set of all measurable functions from S to C (or R) whose absolute value raised to the p-th power has finite integral, or equivalently, that

The set of such functions forms a vector space.

Topology vs Algebra/SigmaAlgebra[edit]

An algebra is a collection of subsets closed under finite unions and intersections. A sigma algebra is a collection closed under countable unions and intersections. In either case, complements are also included.

A topology is a pair (X,Σ) consisting of a set X and a collection Σ of subsets of X, called open sets, satisfying the following three axioms:

  1. The union of open sets is an open set.
  2. The finite intersection of open sets is an open set.
  3. X and the empty set ∅ are open sets. — Preceding unsigned comment added by 128.211.165.166 (talk) 21:12, 26 August 2012 (UTC)

Set cover[edit]

A cover of a set X is a collection of sets whose union contains X as a subset. Formally, if

is an indexed family of sets Uα, then C is a cover of X if

Compact Space[edit]

Formally, a topological space X is called compact if each of its open covers has a finite subcover. Otherwise it is called non-compact. Explicitly, this means that for every arbitrary collection

of open subsets of X such that

there is a finite subset J of A such that