From bd649277c25fae652d6406da78e6eb172029c3de Mon Sep 17 00:00:00 2001 From: Auriel Sierra Date: Sun, 13 Jul 2025 10:00:03 -0600 Subject: [PATCH] trasncripcion --- docs/5_17_1_proceso_funciones_poisson.md | 106 ++++++++++++++++++++++- docs/images/pmf_poisson.png | Bin 0 -> 48358 bytes docs/images/poisson_curvas.png | Bin 0 -> 39852 bytes docs/images/poisson_pasos.png | Bin 0 -> 12002 bytes 4 files changed, 103 insertions(+), 3 deletions(-) create mode 100644 docs/images/pmf_poisson.png create mode 100644 docs/images/poisson_curvas.png create mode 100644 docs/images/poisson_pasos.png diff --git a/docs/5_17_1_proceso_funciones_poisson.md b/docs/5_17_1_proceso_funciones_poisson.md index 48d2628..cceee1b 100644 --- a/docs/5_17_1_proceso_funciones_poisson.md +++ b/docs/5_17_1_proceso_funciones_poisson.md @@ -2,6 +2,106 @@ [17 - Proceso contador de Poisson](https://www.overleaf.com/read/ctgmjmgscrhj#116f35) -### Secciones -- Proceso aleatorio de Poisson (1 - 6) -- Funciones de densidad del proceso de Poisson (7 - 9) \ No newline at end of file +# Proceso aleatorio de Poisson + +!!! note "Introducción" + + Un proceso aleatorio de Poisson describe el número de veces que algún evento ha ocurrido, como una función del tiempo o el espacio, y en donde los eventos ocurren en instantes o lugares al azar. + + Ejemplos de situaciones que pueden describirse con este modelo incluyen: + + - La llegada de un cliente a la caja de supermercado. + - La caída de un rayo dentro de un área prescrita. + - La falla de un componente en un sistema. + - La emisión de un electrón desde la superficie de un material sensible a la luz (fotodetector). + - El número de solicitudes a un servidor. + - El número de plantas en un área boscosa. + + El proceso se reduce a contar el número de tales ocurrencias con el tiempo. Por esta razón, el proceso también se conoce como **proceso contador de Poisson**. + +--- + +## Definición del proceso de Poisson + +!!! tip "Definición del proceso de Poisson" + + Sea \( X(t) \) el número de ocurrencias del evento con el tiempo (el proceso). Entonces \( X(t) \) consiste en funciones de valores enteros no-decrecientes. La probabilidad de que sucedan exactamente \( k \) eventos en el tiempo \( t \) está dada por: + + $$ + P(X = k) = \frac{(\lambda t)^k}{k!} e^{-\lambda t}, \quad k = 0, 1, 2, \ldots + $$ + + Por conveniencia, se toma \( X(t) = 0 \) en \( t = 0 \); para \( t > 0 \), \( X(t) \) es el número de ocurrencias en el intervalo \([0, t]\); para \( t < 0 \), \( X(t) \) es el negativo del número de ocurrencias en el intervalo \([t, 0]\). + + +![Distribución de Poisson para distintos valores de k](images/poisson_curvas.png) + +--- + +!!! tip "Visualización del proceso" + + La evolución del proceso \( X(t) \) puede visualizarse como una función escalonada que aumenta en pasos unitarios en los puntos de ocurrencia del evento. Cada incremento representa un evento observado. + + +![Función muestra del proceso Poisson](images/poisson_pasos.png) + + +--- + +## Condiciones del proceso de Poisson + +!!! tip "Condiciones que debe cumplir el proceso" + + - A la vez solamente ocurre un evento (aunque los tiempos puedan estar muy cerca). + - Los tiempos de ocurrencia son estadísticamente independientes, de modo que el número de eventos en un intervalo dado es independiente del número en otro. + - El número esperado de eventos crece de forma aproximadamente lineal con el tiempo. + + El número de ocurrencias de eventos en cualquier intervalo finito está descrito por la distribución de Poisson con tasa promedio \( \lambda \), también conocida como intensidad. + +--- + +## Ejemplo: Contador de Geiger + +!!! example "¿Cuál es la probabilidad de que en 30 segundos se reciba al menos un pulso?" + + Suponga que en un contador de Geiger llegan pulsos con una tasa promedio de 6 por minuto, de forma que \( \lambda = 6 \). Queremos conocer la probabilidad de que en un intervalo de 30 segundos (es decir, \( t = 0.5 \) minutos) al menos un pulso sea recibido. + + Como el número de pulsos sigue una distribución de Poisson con parámetro \( \lambda t = 3 \), se tiene: + + $$ + P(X \geq 1) = 1 - P(X = 0) = 1 - \frac{(3)^0}{0!} e^{-3} = 1 - e^{-3} \approx 0.95 + $$ + +!!! note "" + Hay un 95 % de probabilidad de detectar al menos un pulso en 30 segundos. + +--- + +# Funciones de densidad del proceso de Poisson + +## Definición de la función de densidad + +!!! tip "Función de densidad del proceso de Poisson" + + La probabilidad de exactamente \( k \) ocurrencias sobre un intervalo \([0, t]\) es: + + $$ + P[X(t) = k] = \frac{(\lambda t)^k}{k!} e^{-\lambda t}, \quad k = 0, 1, 2, \ldots + $$ + + La densidad de probabilidad del número de ocurrencias se puede escribir como: + + $$ + f_X(x) = \sum_{k=0}^{\infty} \frac{(\lambda t)^k}{k!} e^{-\lambda t} \delta(x - k) + $$ + +--- + +!!! tip "Interpretación visual" + + La función de masa de probabilidad (PMF) del proceso Poisson representa la probabilidad de ocurrencia exacta de un número entero de eventos. A medida que el tiempo \( t \) aumenta, la dispersión de la distribución también aumenta, lo que se traduce en una mayor variabilidad del número de eventos esperados. + + +![Función PMF de Poisson con λ = 2.5](images/pmf_poisson.png) + +--- diff --git a/docs/images/pmf_poisson.png b/docs/images/pmf_poisson.png new file mode 100644 index 0000000000000000000000000000000000000000..d61d2ce889fc8b981ba98a88d4815d2b9a3d429f GIT binary patch literal 48358 zcmce;1y@{6(>08{yAzV&Zo%DMg1fuBySqzp2n3g4gF7U+1PeC7U4r|2$aOz!eZSz# zB4>a#)2FMtx@zy;b%<70ltM)!LV|#RK$VdeSA~Fp0z*JR3L(G&zkxUKZ~$H)T~(z- zA!;W{4uBtEEJYMVARrnNk)MoVfu9kbq_tfkAn+*P{zJ;BemsYO==7Bl7g6&vILU)I zQd9pOu_62pk@G5y=xcZ}3=CqGFh=@2y24uV#Sl_`eUVq(V+WN!B2Px6q3=+i>aiFw zl;0tU?u+Q`BXa(dFMsu)T4is`Zu)6v!M5Y;QP;X{ku{#{JL%hfnO@%rhXEt<@2ihm zjr}&{9m2n_I|xkh|Go!{i3cJ7?|XDiNT&bZfFLLi1Ny%=WBmX9W<)+)BeJ(iQ4CrZ z_(zZjK4BUg8*}gFy0W-zOAihWWeec~W_5;Z^}>o^h=v&k+{2oVBuFHZ31EVw5EKWRUz6&pz)o!~?E z-iqf+%FK+EBKQB*shXxFcn`A?_}YbaM9k$dSej`NBgfPxezMYHZ@W$jtcV)Sza@Yi 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