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Sumario Resumen Latex EPE
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Resumen completo a Latex de la asignatura Elementos de Probabilidad y Estadística del grado en Matemáticas en la Universidad de Sevilla
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EPE. José Martı́nez SuárezEPE
Esquema resumen.
Definición
Sea Ω un espacio muestral distinto de vacı́o. Sea {An } una sucesión de sucesos o
subconjuntos de Ω.
Se definen
lı́m sup An = lı́mAn = {ω ∈ Ω : ω es infinidad de An }
n→∞
lı́m ı́nf An = lı́mAn = {ω ∈ Ω : ω ∈ An para casi todo n}
n→∞
Proposición
Sea Ω un espacio muestral distinto de vacı́o. Sea {An } una sucesión de sucesos
contenidos en Ω, entonces se verifica que
1. lı́mAn = ∞
T S∞
n=1 m=n Am
2. lı́mAn = ∞
S T∞
n=1 m=n Am
Proposición
Sea {An } una sucesión de sucesos monótona contenidos en Ω. Entnces se verifica:
1. Si {An } ↑ entonces lı́mAn = lı́mAn = ∞
S
n=1 An .
2. Si {An } ↓ entonces lı́mAn = lı́mAn = ∞
T
n=1 An .
Axiomas de Kolmogorov
Sea (Ω, A ) un espacio probabilizable, definimos la función P : A −→ R+ ∪ {0} que
verifica:
1. ∀A ∈ A , P (A) ≥ 0.
2. Sean A1 , . . . , An , · · · ∈ A dos a dos disjuntos, entonces
∞
! ∞
[ X
P Ai = P (Ai )
i=1 i=1
3. P (Ω) = 1.
Esto genera un espacio probabilı́stico (Ω, A , P ).
Página 1
, EPE. José Martı́nez Suárez
Lema
Se verifican las siguientes propiedades:
1. P (∅) = 0.
2. Sean A, B ∈ A con A ∩ B = ∅, entonces P (A ∪ B) = P (A) + P (B).
3. Sean A, B ∈ A con A ⊆ B, entonces P (A) ≤ P (B).
4. Sea A ∈ A , entonces P (A) = 1 − P (A).
5. Si A, B ∈ A con A ⊂ B, entonces P (B − A) = P (B) − P (A).
6. Sean A, B ∈ A, entonces P (B − A) = P (B) − P (B ∩ A).
7. Sean A, B ∈ A , entonces P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
8. Sean A1 , . . . , An ∈ A, entonces P ( ni=1 An ) ≤ ni=1 P (Ai ).
S P
9. Sean A1 , . . . , An ∈ A , entonces
n
! n
[ X X X
P Aj = P (Aj )− P (Ai ∩ Aj ) + P (Ai ∩ Aj ∩ Ak )
j=1 j=1 i<j i<j<k
n+1
+ · · · + (−1) P (A1 ∩ A2 ∩ · · · ∩ An )
En espacios muestrales finitos la probabilidad se calcula como el cociente
Casos favorables
Casos posibles
Probabilidad condicionada
P (A ∩ B)
P (A|B) =
P (B)
Fórmula de la probabilidad total
∞
X
P (C) = P (C|An ) · P (An )
n=1
Siendo {An } una familia de sucesos disjuntos dos a dos y cuya unión es el total, y C ∈ A .
Fórmula de Bayes. Probabilidad a posteriori
P (Ai ∩ C) P (C|Ai ) · P (Ai )
P (Ai |C) = = P∞
P (C) n=1 P (C|An ) · P (An )
Fórmula de la multiplicación
P (A1 ∩ A2 ∩ · · · ∩ An ) = P (A1 ) · P (A2 |A1 ) · P (A3 |A1 ∩ A2 ) · · · P (An |A1 ∩ · · · ∩ An−1 )
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