User talk:Mindey/MathNotes

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Newton Binomial

${\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}.}$

Notation of Combinations

${\displaystyle C_{n}^{k}={\binom {n}{k}}={\frac {n!}{k!(n-k)!}}}$

A property of Combinations:

${\displaystyle C_{n}^{k}=C_{n}^{n-k}}$

Integral of 1/x

${\displaystyle \int {\frac {1}{x}}dx=ln(x)+C}$

Normal law density and CDF

${\displaystyle X\ \sim \ {\mathcal {N}}(\mu ,\,\sigma ^{2})\Leftrightarrow }$

PDF:

${\displaystyle p_{X}(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}},\quad \ x,\mu ,\sigma \in \mathbb {R} ,\sigma >0.}$

CDF:

${\displaystyle F_{X}(x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)\quad }$, where ${\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt,\quad x\in \mathbb {R} .}$

Continuous r.v. versus Absolutely continuous r.v.

${\displaystyle X}$ is continuous r.v. ${\displaystyle \Leftrightarrow P\{X=x\}=F_{X}(x)-F_{X}(x-0)=0}$

${\displaystyle X}$ is absolutely continous r.v. ${\displaystyle \Leftrightarrow \exists p:\forall x\in \mathbb {R} \ \ F(x)=\int _{-\infty }^{x}p(t)dt}$, or, in discrete case: ${\displaystyle F(x)=\sum _{x_{i}\leqslant x}p_{i}}$

Poisson integral

${\displaystyle \int _{-\infty }^{\infty }e^{-t^{2}}dt={\sqrt {\pi }}}$

Integration by parts heuristic

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

${\displaystyle \int u(x)v'(x)\,dx=u(x)v(x)-\int u'(x)v(x)\,dx}$

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

${\displaystyle P(B\smallsetminus A)=P(B)-P(A\cap B)}$

Definition of Measurable Function = Measurable Mapping ?

Let ${\displaystyle (X,\Sigma )}$ and ${\displaystyle (Y,\mathrm {T} )}$ be measurable spaces, meaning that ${\displaystyle X}$ and ${\displaystyle Y}$ are sets equipped with respective sigma algebras ${\displaystyle \Sigma }$ and ${\displaystyle \mathrm {T} }$. A function

${\displaystyle f:X\to Y}$

is said to be measurable if ${\displaystyle f^{-1}(E)\in \Sigma }$ for every ${\displaystyle E\in \mathrm {T} }$. The notion of measurability depends on the sigma algebras ${\displaystyle \Sigma }$ and ${\displaystyle \mathrm {T} }$. To emphasize this dependency, if ${\displaystyle f:X\to Y}$ is a measurable function, we will write

${\displaystyle f:(X,\Sigma )\to (Y,\mathrm {T} ).}$ — Preceding unsigned comment added by 128.211.164.79 (talk) 02:13, 22 August 2012 (UTC)

Lp space

From undergrad notes: ${\displaystyle l_{p}}$ space, where ${\displaystyle 1\leqslant p\leqslant \infty }$ is a space of sequences, where the distance between the sequences is computed with formula ${\displaystyle d(x,y)={\sqrt[{n}]{\sum _{i=1}^{\infty }|x_{i}-y_{i}|^{p}}}}$. The space will constitute of the sequences with the property ${\displaystyle x=(x_{1},x_{2},...),\quad \sum _{i=1}^{\infty }|x_{i}|^{p}<\infty }$. In other words, this space will be made of sequences, such that their distance from the zero sequence ${\displaystyle (0,0,...)}$ 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

${\displaystyle \|f\|_{p}\equiv \left({\int _{S}|f|^{p}\;\mathrm {d} \mu }\right)^{\frac {1}{p}}<\infty }$

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.

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

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

${\displaystyle C=\lbrace U_{\alpha }:\alpha \in A\rbrace }$

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

${\displaystyle X\subseteq \bigcup _{\alpha \in A}U_{\alpha }}$

Compact Space

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

${\displaystyle \{U_{\alpha }\}_{\alpha \in A}}$

of open subsets of X such that

${\displaystyle X=\bigcup _{\alpha \in A}U_{\alpha },}$

there is a finite subset J of A such that

${\displaystyle X=\bigcup _{i\in J}U_{i}.}$