maths.free › Probability Theory

Probability Theory

Random variables, distributions, expectation, limit theorems.

Lektionen

Introductory Sample spaces and the axioms Kolmogorov's axioms, events as sets, conditional probability and Bayes. 1 - (5/6)^4 Core Random variables and expectation Distributions, expected value, variance, linearity of expectation. 10 * 1/2 Core The common distributions Bernoulli, binomial, Poisson, uniform, exponential, normal, and what each models. integrate e^(-x^2/2)/sqrt(2*pi) dx from -1.96 to 1.96 Advanced The law of large numbers and the central limit theorem Why averages converge and why they are normally distributed. limit of (1 + 1/n)^n as n -> oo

Statistics measures data; probability theory is the mathematics underneath it. Distributions as functions, expectation as an integral, and the laws that make averages settle down.

Mehr Lektionen

Probability

An Introduction to Probability Conditional Probability and Independent Events Bernoulli Trials Discrete Random Variables Central Tendency Probability Spaces with Infinitely Many Outcomes Discussion

Applying Probability to Combinatorics

A First Taste of Ramsey Theory Small Ramsey Numbers Estimating Ramsey Numbers Applying Probability to Ramsey Theory Ramsey's Theorem The Probabilistic Method Discussion

Foundations

Probability theory Probability space Probability axioms Conditional probability Bayes' theorem Independence (probability theory)Random variable Probability distribution Cumulative distribution function Probability density function

Distributions

Binomial distribution Poisson distribution Normal distribution Exponential distribution Uniform distribution (continuous)Geometric distribution Chi-squared distribution Student's t-distribution

Limit theorems

Expected value Variance Covariance Law of large numbers Central limit theorem Chebyshev's inequality Moment-generating function Convergence of random variables Markov chain Martingale (probability theory)Stochastic process Brownian motion

Einige dieser Lektionen sind aus offenen Texten angepasst: Keller & Trotter, Applied Combinatorics, Wikipedia. Jede Unterrichtsseite nennt ihre Quelle und Lizenz.

Hier verwendete Symbole

Tippen Sie auf jedes Symbol für die vollständige Definition, ein Bild und was jeder Buchstabe darin bedeutet.

Fragen, die die Leute stellen

What is the difference between probability and statistics?

Probability goes from a known model to what the data should look like; statistics goes from data back to the model. Probability theory is the deductive half.

What does the law of large numbers promise?

That the average of many independent samples converges to the expected value. It says nothing about any single trial.

Haben Sie eine eigene Aufgabe?
Geben Sie es ein oder zeichnen Sie es auf der Löserseite: jeder Schritt, ein Bild, die Antwort verborgen, bis Sie danach fragen. Lösen →

Weitere Teilgebiete

＋ Arithmetic 𝑥 Algebra △ Geometry ∿ Trigonometry ƒ Precalculus σ Statistics & Probability ⊢ Discrete Math & Logic ∫ Calculus ⊞ Linear Algebra ℕ Number Theory ẏ Differential Equations ∇ Multivariable Calculus ε Real Analysis ⋔ Combinatorics & Graph Theory ∈ Set Theory & Logic 𝔾 Abstract Algebra ≈ Numerical Methods ℂ Complex Analysis ◎ Topology μ Measure Theory ‖·‖ Functional Analysis κ Differential Geometry π₁ Algebraic Topology → Category Theory ∂ Partial Differential Equations ≋ Fluid Dynamics ∞ Frontiers