User talk:Mindey/MathNotes

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Newton Binomial[edit]

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

Notation of Combinations[edit]

C_n^k = \binom nk = \frac{n!}{k!(n-k)!}

A property of Combinations:[edit]

C_n^k = C_n^{n-k}

Integral of 1/x[edit]

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

Normal law density and CDF[edit]

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


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.


    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.[edit]

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[edit]

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

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

\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[edit]

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

Definition of Measurable Function = Measurable Mapping ?[edit]

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 (talk) 02:13, 22 August 2012 (UTC)

Lp space[edit]

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[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 (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

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[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

\{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.