Mathematics

Statistics and linear algebra

Mean, variance, Bayes, matrices, eigenvalues.

Basics

Descriptive stats

The mean is a center of mass; the median is the middle in order; variance measures spread. Standard deviation is in the original units. A normal curve holds about 68% within ±1σ of the mean.

Probability

0≤P≤1, and the whole space sums to 1. Independence: P(A∩B)=P(A)P(B). Conditional: P(A|B)=P(A∩B)/P(B). Bayes reverses the conditioning.

Matrices and vectors

A vector has magnitude and direction; a matrix is a linear map. AB is composition (order matters). Invertible iff det≠0. Eigenvectors are directions the map only stretches.

Formulas

Sample mean and variance

x̄ = (Σ xᵢ)/n, s² = Σ (xᵢ − x̄)² / (n−1)

Sample variance uses n−1 (unbiased). Population variance divides by n.

Symbols

n sample size

Bayes

P(A|B) = P(B|A) P(A) / P(B)

Update a prior P(A) with data B.

Binomial mean

E[X] = n p, Var(X) = n p (1−p)

n independent Bernoulli trials with success p.

Symbols

p success probability

Dot product

a · b = |a| |b| cosθ = Σ aᵢ bᵢ

Zero means perpendicular. Geometry of work W=F·s.

2×2 determinant

det = ad − bc for [[a,b],[c,d]]

Signed area scale. Zero means linearly dependent columns.

Eigenvalues

A v = λ v, det(A − λ I) = 0

Shows up in vibration modes, principal axes, Markov steady states, and more.

Symbols

λ eigenvalue
v eigenvector

Euler's formula

e^{iθ} = cos θ + i sin θ

At θ=π, e^{iπ}+1=0. Language of AC phasors and Fourier analysis.

In this field

Algebra and geometry

Equations, exponentials and logs, Pythagoras, circles.

Trigonometry and calculus

Trig ratios, identities, derivatives, integrals.

Discrete, complex, series

Sequences, series, complex numbers, combinatorics.

Limits and number-theory sketch

Limits, continuity, primes, congruences.

Differential equations and series

First-order linear, harmonic oscillator, Taylor and geometric series.

Vectors and space

Vectors, dot and cross products, planes, and distance in 3D.

Conic sections

Parabola, ellipse, hyperbola, eccentricity, foci, and an orbit sketch.

Fourier sketch

Sines that rebuild a periodic function, coefficients, spectra, and a discrete-transform note.

Probability, continued

Expected value, variance, normal density, and a CLT one-liner.

Graph-theory sketch

Vertices and edges, degree, paths, Euler, and the adjacency matrix.