Distribución multinomial

Multinomial
Parámetros	$n>0$ número de pruebas (entero) $p_{1},\ldots ,p_{k}$ probabilidad de un suceso concreto ( $\Sigma p_{i}=1$ )
Dominio	$X=\left(x_{1},\dots ,x_{k}\right)'\in \mathbb {Z} ^{k}$ $x_{i}\in \{0,\dots ,n\}\ ;\;\Sigma x_{i}=n\!$
Función de densidad (pdf)	${\frac {n!}{x_{1}!\cdots x_{k}!}}p_{1}^{x_{1}}\cdots p_{k}^{x_{k}}$
Media	$E\{X_{i}\}=np_{i}$
Varianza	$\textstyle {\mathrm {Var} }(X_{i})=np_{i}(1-p_{i})$ $\textstyle {\mathrm {Cov} }(X_{i},X_{j})=-np_{i}p_{j}~~(i\neq j)$
Función generadora de momentos (mgf)	${\biggl (}\sum _{i=1}^{k}p_{i}e^{t_{i}}{\biggr )}^{n}$
[editar datos en Wikidata]

En teoría de probabilidad, la distribución multinomial o distribución multinómica es una generalización de la distribución binomial.

La distribución binomial es la probabilidad de un número de éxitos en N sucesos de Bernoulli independientes, con la misma probabilidad de éxito en cada suceso. En una distribución multinomial, el análogo a la distribución de Bernoulli es la distribución categórica, donde cada suceso concluye en únicamente un resultado de un número finito K de los posibles, con probabilidades $p_{1},\dots ,p_{k}$ (tal que $p_{i}\geq 0$ para i entre 1 y K y $\sum _{i=1}^{k}p_{i}=1$ ); y con n sucesos independientes.

Entonces sea la variable aleatoria $X_{i}$ , que indica el número de veces que se ha dado el resultado i sobre los n sucesos. El vector $X=(X_{1},...,X_{k})$ sigue una distribución multinomial con parámetros n y p, donde $p=(p_{1},...,p_{k})$ .

Nótese que en algunos campos las distribuciones categórica y multinomial se encuentran unidas, y es común hablar de una distribución multinomial cuando el término más preciso sería una distribución categórica.

Especificación

Función de probabilidad

La función de probabilidad de la distribución multinomial es como sigue:

{\begin{aligned}f(x_{1},\ldots ,x_{k};n,p_{1},\ldots ,p_{k})&{}=\Pr(X_{1}=x_{1}{\mbox{ y }}\dots {\mbox{ y }}X_{k}=x_{k})\\\\&{}={\begin{cases}{\displaystyle {n! \over x_{1}!\cdots x_{k}!}p_{1}^{x_{1}}\cdots p_{k}^{x_{k}}},\quad &{\mbox{cuando }}\sum _{i=1}^{k}x_{i}=n\\\\0&{\mbox{En otros casos,}}\end{cases}}\end{aligned}}

Para enteros no negativos x₁, ..., x_k.

Propiedades

La esperanza matemática del suceso i observado en n pruebas es:

\operatorname {E} (X_{i})=np_{i}.\,

La varianza es:

\operatorname {var} (X_{i})=np_{i}(1-p_{i}).\,

Datos: Q1147928

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