User talk:Mindey/MathNotes

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Newton Binomial
$$   (x+y)^n = \sum_{k=0}^n {n \choose k}x^{k}y^{n-k}. $$

Notation of Combinations
$$C_n^k = \binom nk = \frac{n!}{k!(n-k)!} $$

A property of Combinations:
$$C_n^k = C_n^{n-k}$$

Integral of 1/x
$$\int \frac{1}{x} dx = ln(x)+ C$$

Normal law density and CDF
$$X\ \sim\ \mathcal{N}(\mu,\,\sigma^2) \Leftrightarrow $$

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

CDF:

F_X(x) = \Phi\left(\frac{x-\mu}{\sigma}\right) \quad $$, where $$ \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.
$$X$$ is continuous r.v. $$\Leftrightarrow P\{X=x\} = F_X(x)-F_X(x-0) = 0$$

$$X$$ is absolutely continous r.v. $$\Leftrightarrow \exists p: \forall x \in \mathbb{R} \ \ F(x) = \int_{-\infty}^x p(t) dt$$, or, in discrete case: $$F(x) = \sum_{x_i \leqslant x} p_i$$

Poisson integral
$$\int_{-\infty}^\infty e^{-t^2} dt = \sqrt{\pi}$$

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


 * $$\int u(x) v'(x) \, dx = u(x) v(x) - \int u'(x) v(x) \, dx $$

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:


 * 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
$$P(B \smallsetminus A) = P(B) - P(A \cap B)$$

Definition of Measurable Function = Measurable Mapping ?
Let $$(X, \Sigma)$$ and $$(Y, \Tau)$$ be measurable spaces, meaning that $$X$$ and $$Y$$ are sets equipped with respective sigma algebras $$\Sigma$$ and $$\Tau$$. A function
 * $$f: X \to Y$$

is said to be measurable if $$f^{-1}(E)\in \Sigma$$ for every $$E\in\Tau$$. The notion of measurability depends on the sigma algebras $$\Sigma$$ and $$\Tau$$. To emphasize this dependency, if $$f : X\to Y$$ is a measurable function, we will write
 * $$f: (X,\Sigma)\to (Y,\Tau).$$ — Preceding unsigned comment added by 128.211.164.79 (talk) 02:13, 22 August 2012 (UTC)

Lp space
From undergrad notes: $$l_p$$ space, where $$1 \leqslant p \leqslant \infty$$ is a space of sequences, where the distance between the sequences is computed with formula $$d(x,y) = \sqrt[n]{\sum_{i=1}^{\infty} | x_i - y_i |^{p}}$$. The space will constitute of the sequences with the property $$x=(x_1, x_2, ...), \quad \sum_{i=1}^{\infty} | x_i |^p < \infty$$. In other words, this space will be made of sequences, such that their distance from the zero sequence $$(0,0,...)$$ 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
 * $$\|f\|_p \equiv \left({\int_S |f|^p\;\mathrm{d}\mu}\right)^{\frac{1}{p}}<\infty$$

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
 * $$C = \lbrace U_\alpha: \alpha \in A\rbrace$$

is an indexed family of sets Uα, then C is a cover of X if
 * $$X \subseteq \bigcup_{\alpha \in A}U_{\alpha}$$

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


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

of open subsets of $X$ such that


 * $$X=\bigcup_{\alpha\in A} U_\alpha,$$

there is a finite subset $J$ of $A$ such that


 * $$X=\bigcup_{i\in J} U_i.$$