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Unidad1

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Tema 1, Estadistica, Ingenieria Telemática, UPCT

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Unidad1

  1. 1. Fundamentos de la Teor´ıa de la probabilidad Pilar Sanmart´ın. departamento de matem´atica Aplicad y Estad´ıstica. Universidad Polit´ecnica de Cartagena Estad´ıstica GIST, GIT Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  2. 2. Introducci´on Concepto presente en la vida cotidiana. EJEMPLOS Es probable que llueva. Es muy poco probable que pase la prueba. Es casi imposible que la estructura resista. OBJECTIVO Objectivo: Cuantificar la veracidad (verosimilitud) de hechos (sucesos) inciertos. Probabilidad Matem´atica: Cuantificar num´ericamente la verosimilitud de sucesos inciertos. Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  3. 3. Incertidumbre? Cressie, N. Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  4. 4. Incertidumbre? Cressie, N. Uncertainty is everywhere; as the famous quote by Benjamin Franklin (Sparks, 1840) says “ In this world the only certainties are death and taxes. ”Not only our world is un- certain, but also our attempts to explain it are (i.e. science). And our measures of our world (uncertain) are uncertain.” Statistics is the science of uncertainty and it offers a coherent approach to dealing with all the above sources of uncertainty. Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  5. 5. INTRODUCCION INCERTIDUMBRE C´omo llegar a resultados concluyentes? incorporando la incertidumbre. T´eor´ıa de la Probabilidad Teor´ıa Matem´atica que puede modelizar experimentos cuyo resultado es imposible de predecir exactamente (Experimentos aleatorios). Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  6. 6. Incertidumbre Teor´ıa de colas Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  7. 7. Razonamiento bayesiano y aprendizaje m´aquina Conferencia:Dr. Figueiras Titulo: Inferencia M´aquina y Aprendizaje Profundo (una concisa introduccin) El Aprendizaje Profundo (Deep Learning) se puede considerar una de las t´ecnicas de aprendizaje artificial m´as potentes desarrolladas de esta ´ultima d´ecada. Es considerada como una tecnolog´ıa clave en el futuro de la inteligencia artificial y actualmente es utilizada en casi todos los campos de la ciencia: procesado de datos, voz, imagen, visi´on artificial, etc. En esta charla, el Dr. Figueiras realizar´a una breve introducci´on al aprendizaje profundo desde el punto de vista del aprendizaje m´aquina, abordando aspectos tan relevantes como la diversidad y el Big Data. Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  8. 8. INTRODUCCION INCERTIDUMBRE C´omo llegar a resultados concluyentes? incorporando la incertidumbre. T´eor´ıa de la Probabilidad Teor´ıa Matem´atica que puede modelizar experimentos cuyo resultado es imposible de predecir exactamente (Experimentos aleatorios). Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  9. 9. Contenidos 1 Conceptos b´asicos relacionados con un experiemnto. 1 Definiciones. 2 Operaciones elementales con sucesos. 2 El concepto de Probabilidad. 1 Definici´on Axiom´atica de la Probabilidad. 2 espacios muestrales finitos. 3 Probabilidad Condicionada. 1 Definici´on. 2 Formula de la probabilidad Total. Teorema de Bayes. 4 Independencia. Fiabilidad. Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  10. 10. Conceptos b´asicos relacionados con un experimento EXPERIMENTO: cada acci´on o proceso que genera observaciones. Prueba planeada en un laboratorio . lanzar un dado . EXPERIMENTO ALEATORIO: aquel que produce resultados variables que no son conocidos con certeza previo a la realizaci´on del experimento. Consta de: Procedimiento: lanzar una moneda. Observacion: Observar que lado (cara o cruz) queda a la vista. Modelo: Caras y cruces son igualmente probables. El resultado de cada lanzamiento no est´a relacionado con los resultados de los lanzamientos previos. Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  11. 11. Conceptos b´asicos relacionados con un experimento la base matem´atica de la probabilidad es la teor´ıa de conjuntos. (Espacio muestral como conjunto Universal, resultados como elementos, sucesos como subconjuntos) Resultado: un resultado es cada una de las posibles observaciones del experimento. ESPACIO MUESTRAL (S) el conjunto de todos los resultados del experimento. S = {s1, . . . , sn} EVENT (E): cada colecci´on (subconjunto) de resultados contenidos en el espacio muestral S. Cardinal, |E| = n´umero de elementos(resultados) Simple. |E| = 1 Compuesto.|E| > 1 Es usual recurrir a los disgramas de Venn para mostrar las relaciones entre los subconjuntos (sucesos). SUCESO SEGURO: S SUCESO IMPOSIBLE (conjunto nulo) : ∅ Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  12. 12. Conceptos b´asicos relacionados con un experimento La union de dos sucesos A y B,A ∪ B, es el suceso consistente en todos los resultados que est´an en A o B. A ∪ B = {s|s ∈ Ao s ∈ B} la intersecci´on de dos sucesos A y B, A ∩ B, es el suceso pformado por todos los resultados que estan en A y B. A ∩ B = {s|s ∈ A y s ∈ B} El suceso complementario de A, Ac (A), es el conjunto de resultados de S que no pertenecen a A. Ac = {s|s /∈ A} Suceso disjuntoss: cuando no tienen elementos en com´un A ∩ B = ∅ Leyes de Morgan: (A ∪ B)c = Ac ∩ Bc (A ∩ B)c = Ac ∪ Bc A ⊆ B ∀s ∈ A −→ s ∈ B Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  13. 13. Conceptos b´asicos relacionados con un experimento Propiedad conmutativa de la uni´on y la intersecci´on: A ∪ B = B ∪ A A ∩ B = B ∩ A Propiedad asociativa de la uni´on y la intersecci´on: (A ∪ B) ∪ C = A ∪ (B ∪ C) = (A ∪ C) ∪ B (A ∩ B) ∩ C = A ∩ (B ∩ C) = (A ∩ C) ∩ B Propiedad Distributiva : (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C) (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C) Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  14. 14. Conceptos b´asicos relacionados con un experimento Espacio de Sucesos ( tambi´en partici´on de S) {Ei }i∈I , es una colecci´on de sucesos colectivamente exhaustiva, mutuamente exclusiva. mutuamente exclusiva. Ei ∪ Ej = ∅ i = j colectivamente exhaustiva i∈I Ei = S Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  15. 15. Concepto de probabilidad. Interpretaciones de la probabilidad Cl´asica: desarrollado originalmente en relaci´on con los juegos de azar. Los sucesos inciertos relevantes se describen como colecciones de resultados elementales que se consideran igualmente ”creibles”. P(A) = casos favorables a A casos posibles Frecuentista: aparece en el contexto de experimentos reproducibles, como los de laboratorio.Los sucesos inciertos relevantes se definen en situaciones experimentales controladas cuyas condiciones de realizaci´on se pueden repetir una y otra vez, indefinidamente. P(A) = ocurrencias de A ensayos realizados l´ımite en la serie de repeticiones del experimento. Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  16. 16. Interpretations of probability. Probability Concept SUBJECTIVA: Nuestro conocimiento sobre la ocurrencia de un suceso A, a priori antes de realizar el experimento P(A) . (prior) a objetivo com´un: medir la verosimilitud de la ocurrencia de sucesos inciertos. Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  17. 17. Concepto de Probabilidad Definici´on axiom´atica Una distribuci´on de probabilidad P[.] es una funci´on que asigna n´umeros reales a sucesos del espacio muestral de forma que se cumplen los siguientes axiomas: Axiom 1 para cada suceso A de inter´es , P(A) ≥ 0 Axiom 2 P(S) = 1 Axiom 3 para cada colecci´on numerable {Ai }i∈I de sucesos mutuamente excluyentes (Ai ∩ Aj = ∅ ∀i = j), P( i Ai ) = i P(Ai ) Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  18. 18. Propiedades Como consecuencia del Axiom 3 se cumple P(A1 ∪ A2) = P(A1) + P(A2) A1 and A2 mutually exclusive. Como consecuencia del Axiom 3 lse cumple el caso particular I = {1, 2, . . ., n} B = {s1, . . . , sm} P(B) = m i=1 P({si }) Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  19. 19. Espacios muestrales finitos Consideramos experimentos para los cuales hay un conjunto finito de resultados posibles |S| = n. S = {s1, . . . , sn} pi = P({si }). Una distribuci´on sobre S queda completamente especificada por {pi }n i=1, cumpliendo: 1 0 ≤ pi ≤ 1 ∀i 2 i pi = 1 Se cumple ∀A: P(A) = si ∈A pi Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  20. 20. resultados equiprobables (Espacios Muestrales Simples) |S| = n p = P({si }). En este caso: p = 1/|S| Se cumple : P(A) = casos favorables a A casos posibles = |A| |S| (1) Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  21. 21. Propiedades la medida de probabilidad cumple: P(∅) = 0. P(Ac ) = 1 − P(A) Para cada A y B (no necesariamente disjuntos) P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Si A ⊂ B P(A) ≤ P(B). Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  22. 22. Probabilidad Condicionada Queremos actualizar nuestras asignaciones a priori P(A) de probabilidades asociadas con sucesos de un experimento aleatorio, basandonos en la llegada de nueva informaci´on. En particular en el conocimiento de que un nuevo suceso B ha ocurrido pero no sabemos el resultado preciso de B. En este contexto, calculamos P(A|B) ”la probabilidad de A dado B”: Probabilidades condicionadas. Sea B ⊆ Ω tal que P(B) > 0. La probabilidad condicionada dado B, es una asignaci´on de n´umeros P(A|B) donde: P(A|B) = P(A ∩ B) P(B) P(A|B) cumple los axiomas de la definici´on de probabilidad. Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  23. 23. Regla del producto Se aplica para calcular la probabilidad asociada a la interseci´on de dos sucesos: P(A ∩ B) = P(A|B)P(B) De tres: P(A ∩ B ∩ C) = P(A|B ∩ C)P(B|C)P(C) En general ... P(∩n i=1Ai ) = P(A1| ∩n i=2 Ai )P(A2| ∩n i=3 Ai ) . . . P(An−1|An)P(An) Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  24. 24. Ley de la probabilidad total Sea {Ai }n i=1 ) partici´on of S, (Espacio de Sucesos) y B un suceso de inter´es. Supongamos P(Ai ) y P(B|Ai ) i = 1, . . . , n conocidos. Se cumple que: P(B) = n i=1 P(B|Ai )P(Ai ) Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  25. 25. Teorema de Bayes Sea {Ai }n i=1 partici´on de S, B suceso de inter´es, P(Ai ) y P(B|Ai ) i = 1, . . . , n conocidos. para cada Ai se cumple: P(Ai |B) = P(B|Ai )P(Ai ) n j=1 P(B|Aj )P(Aj ) Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  26. 26. Independencia de sucesos Definici´on A y B independientes ↔ P(A ∩ B) = P(A)P(B). cuando A y B htienen probabilidades no nulas es equivalente a P(A|B) = P(A) P(B|A) = P(A) El conocimiento de que B ha ocurrido no cambia la probabilidad de ocurrencia del suceso. Independencia de n sucesos {Ai }n i=1 son independientes si para cada subfamilia {Aij }k j=1 se cumple: P(∩k j=1Aij ) = k j=1 P(Aij ) Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  27. 27. Problemas de Fiabilidad Pruebas independientes: Experimentos consistentes en repeticiones independientes de un subexperimento. Las pruebas independientes pueden utilizarse para describir problemas de fiabilidad en los cuales nos gustar´ıa calcular la probabilidad de que una operaci´on particular tenga exito. La operaci´on consta de n componenetes. Cada componente funciona Wi con probabilidad p, independientemente de cualquier otra componente. Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  28. 28. Problemas de fiabilidad Dos tipos de operaciones b´asicas: Componentes en serie: la operaci´on progresa si todas las componentes funcionan. P(W ) = P(∩Wi ) = pn Componentes en paralelo: la operaci´on progresa si alguna componente funciona. P(W ) = 1 − P(W c ) = 1 − (1 − p)n Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad
  29. 29. Problemas de fiabilidad Pilar Sanmart´ın. Fundamentos de la Teor´ıa de la probabilidad

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