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Asignatura: Statistics I, Profesor: , Carrera: Administració i Direcció d'Empreses - Anglès, Universidad: UAB
Tipo: Apuntes
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¡No te pierdas las partes importantes!
The 8 properties below are derived from the 3 axioms by Kolmogorov, and are very useful to compute the probability of any event.
P(ωi) =
n
(i = 1 , 2 ,... , n)
P(A) =
nA n
(i = 1 , 2 ,... , n)
⋃^ m
i= 1
Ai) =
m
i= 1
P(Ai) Ai ∩ A (^) j = 0 for/ i 6 = j
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Exemple 24 A dice is tossed. We are interested in the events A =“The outcome is an even number”, B =“The outcome is less than 3”, C =“The outcome is not 6” and D =“The outcome is 1” (respectively, A = { 2 , 4 , 6 } B = { 1 , 2 } C = { 1 , 2 , 3 , 4 , 5 } D = { 1 } ). We will have:
Furthermore, we might compute other probabilities such as:
P(”The outcome is 3 or more than 3”) = P( B¯) = 1 − P(B) = 1 −
P(”The outcome is an even numner or 1”) = P(A ∪ D) = P(A) + P(D) =
= =
P(”The outcome is an even number or is not 6”) = P(A ∪C) = P(A) + P(C) −
−P(A ∩C) = = P({ 2 , 4 , 6 }) + P({ 1 , 2 , 3 , 4 , 5 }) − −P({ 2 , 4 }) =
=
Definició 25 Let Ω be the sample space associated to a random experiment and let A the corresponding σ -àlgebra of events. Let B ∈ A be an event such that P(B) > 0. Then, for any event A ∈ A , we say that the probability of A conditional on B is: P(A/B) =
From the conditional probability formula we get:
We can also verify that:
Notice that when A and B are independent,
As a matter of fact, this “multiplication rule”, which is true if and only if A and B are independents, is what is used to define the concept of stochastic independence
Definició 27 Two events A, B ∈ A are stochastically independent or, simply, independent if: P(A ∩ B) = P(A) · P(B)
The concept of independence generalizes to 3 or more events in the following way:
Definició 28 A set of n events A 1 ,... , An ∈ A are stochastically independent or, simply, independent if for any subset of 2 or more events we have that:
P(Ak 1 ∩ Ak 2 ∩... ∩ Akm) = P(Ak 1 ) · P(Ak 2 ) ·... · P(Akm)
Exemple 29 A dice is tossed. Consider the events:
Intuitively, we have that A=”The outcome is an odd number”, B=”The out- come is an even number” i C=”The outcome is less that 5”.
Clearly A and B ARE NOT independent since the occurrence of any of them implies that the other cannot occur. Thus, the occurrence of one of them yields a lot of information about the probability that the other occurs.
On the other hand, C IS independent both from A and from B. Indeed, if we are told that C has occurred then we know that the outcome is either an even number (2 or 4) or an odd number (1 or 3), with the same probability. Thus we are not getting any relevant information regarding the probability that the outcome is an odd number (event A) or and odd number (event B)
The following properties are important regarding the concept of independence:
We are going to see two important theorems in probability theory. They have important applications in the use and analysis of information.
2.6.1 Total probability theorem
This theorem states that the probability of any event can be computed using the information about other “small” events.
Theorem 30 Let A be a σ -algebra of events and A 1 , A 2 ,... , An a set of events such that:
(i) Ai ∩ A (^) j = 0 / for i 6 = j.
(ii) A 1 ∪ A 2 ∪... ∪ An = Ω
((i) and (ii) mean A 1 , A 2 ,... , An form a partition of Ω)
(iii) P(Ai) > 0 ∀Ai (i = 1 ,... , n)
Then, for any B ∈ A :
n
i= 1
P(B/Ai) · P(Ai)
