Sebaran probabilitas

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Dina matematika, sebaran probabilitas nangtukeun unggal interval tina wilangan nyata kamungkinan, mangka kitu aksioma probabilitas terpenuhi. Dina watesan teknik, probabiliti sebaran nyaeta ukuran probabilitas numana domain mangrupa aljabar Borel dina kaayaan riil.

Probabilitas sebaran dina kasus husus ngarupakeun notasi nu leuwih tina ukuran probabilitas, nyaéta fungsi nu assigns probabilities satisfying the Kolmogorov axioms to the measurable sets of a measurable space.

Unggal variabel acak gives rise to a probability distribution, and this distribution contains most of the important information about the variable. If X is a random variable, the corresponding probability distribution assigns to the interval [a, b] the probability Pr[a ≤ X ≤ b], i.e. the probability that the variable X will take a value in the interval [a, b].

Sebaran probabilitas variabel X bisa sacara unik didadarkeun ku fungsi sebaran kumulatif F(x), nu ditangtukeun ku

$F(x) = {\rm Pr} \left[ X \le x \right]$

pikeun x anggota R.

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A distribution is called discrete if its cumulative distribution function consists of a sequence of finite jumps, which means that it belongs to a discrete random variable X: a variable which can only attain values from a certain finite or countable set. A distribution is called continuous if its cumulative distribution function is continuous, which means that it belongs to a random variable X for which Pr[ X = x ] = 0 for all x in R.

The so-called absolutely continuous distributions can be expressed by a fungsi dénsitas probabilitas: a non-negative Lebesgue integrable function f defined on the reals such that

${\rm Pr} \left[ a \le X \le b \right] = \int_a^b f(x)\,dx$

for all a and b. That discrete distributions do not admit such a density is unsurprising, but there are continuous distributions like the devil's staircase that also do not admit a density.

The support of a distribution is the smallest closed set whose complement has probability zero.

[édit] Jejer penting dina sebaran probabiliti

Sababaraha sebaran probabiliti kacida pentingna dina teori atawa pamakean dibere ngaran nu husus:

Sebaran diskrit
- Dina kaayaan terhingga
  - Sebaran degenerate dina x₀, numana X ngarupakeun nilai penting dicokot jadi nilai x₀. This does not look random, but it satisfies the definition of variabel acak. This is useful because it puts deterministic variables and random variables in the same formalism.
  - The discrete uniform distribution, where all elements of a finite set are equally likely. This is supposed to be the distribution of a balanced coin, an unbiased die, a casino roulette or a well-shuffled deck. Also, one can use measurements of quantum states to generate uniform random variables. All these are "physical" or "mechanical" devices, subject to design flaws or perturbations, so the uniform distribution is only an approximation of their behaviour. In digital computers, pseudo-random number generators are used to produced a statistically random discrete uniform distribution.
  - The Bernoulli distribution, which takes value 1 with probability p and value 0 with probability q=1-p.
  - Sebaran binomial, nu ngajelaskeun jumlah kasuksesan dina deret tina percobaan bebas Enya/Henteu.
  - The hypergeometric distribution, which describes the number of successes in the first m of a series of n independent Yes/No experiments, if the total number of successes is known.
- With infinite support
  - The geometric distribution, a discrete distribution which describes the number of attempts needed to get the first success in a series of independent Yes/No experiments.
  - The negative binomial distribution, a generalization of the geometric distribution to the nth success.
  - The Poisson distribution, which describes the number of rare events that happen in a certain time interval.
  - The Boltzmann distribution, a discrete distribution important in statistical physics which describes the probabilities of the various discrete energy levels of a system in thermal equilibrium. It has a contiuous analogue. Special cases include
    - The Gibbs distribution
    - The Maxwell-Boltzmann distribution
    - The Bose-Einstein distribution
    - The Fermi-Dirac distribution
  - The zeta distribution has uses in applied statistics and statistical mechanics, and perhaps may be of interest to number theorists.
Continuous distributions
- Supported on a finite interval
  - sebaran seragam dina [a,b], numana sakabeh titik dina interval nu kawengku mibanda ajen nu ampir sarua.
  - Sebaran beta dina [0,1], ngarupakeun sebaran seragam dina kasus husus, nu dipake dina estimasi probabiliti sukses.
  - The Triangular distribution on [a, b]
- Supported on semi-infinite intervals, usually [0,∞)
  - Sebaran eksponensial, which describes the time between consecutive rare random events in a process with no memory.
  - Sebaran gamma, nu ngajelaskeun waktu salila n consecutive rare random events occur in a process with no memory.
  - The Erlang distribution, which is a special case of the gamma distribution with integral shape parameter, developed to predict waiting times in queuing systems.
  - The Log-normal distribution, describing variables which can be modelled as the product of many small independent positive variables.
  - The Weibull distribution, of which the exponential distribution is a special case, is used to model the lifetime of technical devices.
  - Sebaran chi-kuadrat, nu ngarupakeun jumlah kuadrat n variabel random Gauss bebas. Ieu ngarupakeun kasus husus dina sebaran Gamma, sarta dipake dina tes goodness-of-fit dina statistik.
  - Sebaran-F, numana sebaran rasio dua sebaran variabel normal, dipake dina analisa varian.
- Supported on the whole real line
  - Sebaran normal, disebut oge Gaussian atawa kurva bel. It is ubiquitous in nature and statistics due to the central limit theorem: every variable that can be modelled as a sum of many small independent variables is approximately normal.
  - Sebaran-t student, dipake keur nga-estimasi mean nu teu dipikanyaho dina populasi Gaussian.
  - Sebaran Cauchy, conto sebaran nu teu ngabogaan nilai ekspektasi atawa varian. Dina fisika biasana disebut Lorentzian, sarta ieu sebaran tina tetapan energi teu stabil dina mekanika kuantum. Dina fisika partikel, the extremely short-lived particles associated to such unstable states are called resonances.
Joint distributions
- Two or more random variables on the same sample space
  - Bivariate distribution
  - Conditional distribution
  - Multivariate distribution
  - Multinomial distribution, a generalization of the sebaran binomial.
Sebaran nilai-matrix
- sebaran Wishart
- Matrix normal distribution
- Matrix T distribution
- Normal-inverse Wishart distribution