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 '(x) dx and dv = v'(x) dx, 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:
- The union of open sets is an open set.
- The finite intersection of open sets is an open set.
- 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
- ^ Kasube, Herbert E. (1983). "A Technique for Integration by Parts". The American Mathematical Monthly. 90 (3): 210–211. doi:10.2307/2975556. JSTOR 2975556.