- 1 Newton Binomial
- 2 Notation of Combinations
- 3 A property of Combinations:
- 4 Integral of 1/x
- 5 Normal law density and CDF
- 6 Continuous r.v. versus Absolutely continuous r.v.
- 7 Poisson integral
- 8 Integration by parts heuristic
- 9 Probability of difference of events
- 10 Definition of Measurable Function = Measurable Mapping ?
- 11 Lp space
- 12 Topology vs Algebra/SigmaAlgebra
- 13 Set cover
- 14 Compact Space
Notation of Combinations
A property of Combinations:
Integral of 1/x
Normal law density and CDF
- , where
Continuous r.v. versus Absolutely continuous r.v.
is continuous r.v.
is absolutely continous r.v. , or, in discrete case:
Integration by parts heuristic
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
- 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
Definition of Measurable Function = Measurable Mapping ?
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
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
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.
- 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 126.96.36.199 (talk) 21:12, 26 August 2012 (UTC)
is an indexed family of sets Uα, then C is a cover of X if
of open subsets of X such that
there is a finite subset J of A such that