Distribución xeométrica

Para outras páxinas con títulos homónimos véxase: Distribución.

A distribución xeométrica é calquera das dúas distribucións de probabilidade discretas seguintes:

a distribución de probabilidade do número X do ensaio de Bernoulli necesaria para obter un éxito, contido no conxunto { 1, 2, 3,...} ou
a distribución de probabilidade do número Y = X − 1 de fallos antes do primeiro éxito, contido no conxunto { 0, 1, 2, 3,... }.

Propiedades

A distribución xeométrica non ten memoria, é dicir, $P(X>m+n|X>m)=P(X>n),$ . Isto significa que se intentamos repetir o experimento ata o primeiro éxito, entón, dado que o primeiro éxito aínda non ocorreu, a distribución da variable condicionada do número de ensaios adicionais non depende de cantos fallos se observaron. De feito, a distribución xeométrica é a única distribución discreta sen memoria.
Se a probabilidade de éxito en cada ensaio é p, entón a de que sexan necesarios x para obter un éxito é $P(X=x)=(1-p)^{x-1}p\,$ para x = 1, 2, 3,.... Equivalentemente, a probabilidade de que haxa x fallos antes do primeiro éxito é $P(Y=x)=(1-p)^{x}p\,$ para y = 0, 1, 2,... En ambos os casos, a secuencia é unha progresión xeométrica.
O valor esperado dunha variable aleatoria X distribuída xeometricamente é $\ E(X)={\frac {1}{p}}$ e dado que Y = X-1, $\ E(Y)={\frac {1-p}{p}}$ .
En ambos os casos, a varianza é ${\mbox{var}}(Y)={\mbox{var}}(X)={\frac {1-p}{p^{2}}}$ .
As funcións xeratrices de X e a de Y son, respectivamente, $G_{X}(s)={\frac {sp}{1-s(1-p)}}\quad {\textrm {y}}\quad G_{Y}(s)={\frac {p}{1-s(1-p)}},\quad |s|<(1-p)^{-1}$ .
De todas estas distribucións contidas en {1, 2, 3,... } cun valor esperado dado μ, a distribución xeométrica X con parámetro p = 1/μ é a de maior entropía.
A distribución xeométrica do número y de fallos antes do primeiro éxito é infinitamente divisible, é dicir, para calquera enteiro positivo n, existen variables aleatorias independentes Y ₁,..., Y_n distribuídas identicamente a suma das que teñen a mesma distribución que ten Y. Estas non serán xeométricamente distribuídas agás se n = 1.

Distribucións relacionadas

A distribución xeométrica é un caso especial da distribución binomial negativa con parámetro k = 1. Máis generalmente, se Y ₁,...,Y_k son variables independentes distribuídas xeometricamente con parámetro p, entón $Z=\sum _{m=1}^{k}Y_{m}$ segue unha distribución binomial negativa con parámetros k e p.

Se Y₁,...,Y_r son variables independentes distribuídas xeometricamente (con diferentes parámetros de éxito p_m posibles ), entón o seu mínimo $W=\min _{m}Y_{m}$ segue tamén unha distribución xeométrica, con parámetro

$p=1-\prod _{m}(1-p_{m})$ .

Véxase tamén

Ligazóns externas

Calculadora Distribución xeométrica

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