Discrete Stochastics

Bearbeitet von Otto Moeschlin, Eugen Grycko, Carsten Poppinga, Frank Steinert, P Bäumle-Courth

1. Auflage 2003. CD. VIII, 104 S. ISBN 978 3 540 14913 2 Format (B x L): 15,5 x 23,5 cm Gewicht: 177 g

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In this chapter we introduce the term discrete . Discrete proba- bility spaces are characterized by a finite or countably infinite, i.e. discrete ba- sic space. The main concept is the probability for which σ-additivity is demanded. With help of the probability a probability measure can uniquely be defined on a discrete basic space.

With the motivations given in Chapter 3 as a background we present the definition of a discrete probability space.

4.1 Definition The pair (Ω,P ) is called a discrete probability space, if Ω is a non-empty, finite or countably infinite (i.e. a discrete) , and P : P(Ω) → R is a mapping from the P(Ω) of Ω to the real numbers with the following properties: (4.1.1) P (A) ≥ 0 (A ⊂ Ω) (Non-negativity) (4.1.2) P (Ω) = 1 (Normalization)

(4.1.3) for every (An) of pairwise disjoint sets from P(Ω), X  X P An = P (An)(σ-additivity) n∈N n∈N (read: sigma-additivity) Ω is called the basic space or the sample space, P a (discrete) prob- ability measure, where, if we want to be more precise, the basic space will be mentioned too, i.e.: (discrete) probability measure over Ω. In the 0 following, Ω will be either Z or such suitable subsets of Z as N, Nn, N or 0 Nn, or cartesian products of these sets, although parts of the theory can be formulated for abstract discrete basic spaces.

4.2 Remarks 4.2.1 In the following, the adjective “discrete” will be omitted if it is not required for emphasis or differentiation. 4.2.2 In the sequel the set Ω is called the basic space only. 20 4 Discrete Probability Spaces

4.2.3 The subsets of the basic space Ω are called events. P (A) is understood as the probability of the event A or – in the language of the random experiment which serves as an intuitive background – as the probability that a randomly selected element ω ∈ Ω belongs to the set A, i.e. ω ∈ A. 4.2.4 Note that P is not a mapping of the set Ω but of the set P(Ω) to R; i.e. it is the subsets (not the points) of Ω to which the real numbers are assigned. Therefore P is called a set function. 4.2.5 If for A ∈ P(Ω) either

P (A) = 0 or P (A) = 1,

A is called a (P-)null set or a (P -)one set, respectively. ∅ is always a null set, and Ω is a one set.

At the moment we are content with a first example of a probability mea- sure, the point probability measure. This provides typical examples for non- trivial null and one sets.

4.3 Example (Point probability measure)

Let ω ∈ Ω. Then δω : P(Ω) → [0; 1] with ( 1, if ω ∈ A, δω(A) := (A ⊂ Ω) 0, if ω∈ / A is called the point probability measure in ω; this is also referred to as the Dirac-measure in ω. Evidently the whole mass is concentrated in ω. If ω ∈ A for A ⊂ Ω, then δω(A) has the value 1, i.e. A is a one set (under this point probability measure). If ω∈ / A then δω(A) becomes 0 and A is a null set (under this point probability measure). Convince yourself that δω is a probability measure; especially consider the σ-additivity.

4.4 Corollary 4.4.1 P (∅) = 0,

4.4.2 P (A + B) = P (A) + P (B)(A, B ⊂ Ω,A ∩ B = ∅), (Additivity of P )

4.4.3 P (Ac) = 1 − P (A)(A ⊂ Ω),

4.4.4 P (B \ A) = P (B) − P (A)(A, B ⊂ Ω,A ⊂ B), 4.6 Definition 21

4.4.5 A ⊂ B =⇒ P (A) ≤ P (B)(A, B ⊂ Ω)

4.4.6 P (A) ≤ 1 (A ⊂ Ω).

Proof: 4.4.1: Because of (4.1.2) and (4.1.3) we have 1 = P (Ω) = P (Ω + ∅ + ··· ) = P (Ω) + ∞ · P (∅), which implies ∞ · P (∅) = 0; together with (1.5.3) we have P (∅) = 0. 4.4.2: By (4.1.3), 4.4.1, and (1.5.3), P (A+B) = P (A+B+∅+··· ) = P (A)+P (B)+0+··· = P (A)+P (B). (Evidently in the case of a finite number of non-empty sets σ- additivity of P implies the additivity of P .) 4.4.3: By (4.1.2) and 4.4.2 we have 1 = P (Ω) = P (A + Ac) = P (A) + P (Ac). 4.4.4: For A ⊂ B, we have B = (B \ A) + A and therefore, by 4.4.2, P (B) = P (B \ A) + P (A). 4.4.5: This is a consequence of 4.4.4 together with (4.1.1). 4.4.6: This result is a special case of 4.4.5 with B := Ω. 

4.5 Remark The probability measure has been introduced as a mapping P : P(Ω) → R, i.e. it assigns a to every subset of Ω. But the probability measure P is already determined if the proba- bility values P { ω } are known for all ω ∈ Ω. From the σ-additivity of P and the countability of Ω it follows that for all A ⊂ Ω  X  X  (4.5.1) P (A) = P { ω } = P { ω } . ω∈A ω∈A (Note that the sum in the right hand term of (4.5.1) can consist of either a finite or a countably infinite sum; also see the proof of 4.4.2.)

4.6 Definition Let Ω be a non-empty, discrete set. 4.6.1 A function w : Ω → [0; 1] is called probability function (on Ω), P if ω∈Ω w(ω) = 1. In the case that Ω is countable, summation means to calculate the value of the respective non-negative series. 22 4 Discrete Probability Spaces

4.6.2 Let P be a probability measure over Ω. Then the mapping w : Ω → [0; 1] defined by

w(ω) := P { ω } (ω ∈ Ω)

is called the probability function of P .

4.7 Theorem A probability function w : Ω → [0; 1] uniquely defines a probability measure P over Ω, and X (4.7.1) P (A) = w(ω)(A ⊂ Ω). ω∈A

Proof: We can restrict the proof to establishing the σ-additivity of P , particularly since the other two properties of a probability measure, the non-negativity and the normalization, are fulfilled already. Let (An) be a sequence of pairwise disjoint sets in Ω. Then the permuta- tion theorem for series with non-negative terms implies that

 X  X X X X P An = w(ω) = w(ω) = P (An). P n∈N ω∈ An n∈N ω∈An n∈N In particular, from (4.7.1) and the permutation theorem for series with non- negative terms it follows that P (A) is uniquely determined for all A ⊂ Ω, i.e. it does not depend on the order of summation. 

4.8 Discrete probability measures over uncountable basic spaces

Occasionally it is convenient to consider a discrete probability measure over an uncountable basic space, e.g. over R. This leads, for instance in connection with probability functions, to the problem of a summation with uncountably many terms. If P is a discrete probability measure over an uncountable basic space Ω0, this simply means that there exists a countable set Ω, Ω ⊂ Ω0, as an basic space over which P is defined. Within a summation process we take into account – even without ex- plicit mention – only those terms that are indicated by elements of Ω; terms that are indicated by elements of Ω0 \Ω are omitted. Of course, the set Ω can be replaced by any convenient countable superset D ⊂ Ω0 of Ω.

S 4.9 Exercise Let (Ω,P ) be a probability space and B ⊂ Ω with P (B) > 0. Show that 4.10 Exercise 23 P (A ∩ B) P (A | B) := (A ⊂ Ω) P (B) defines a probability measure P ( . | B) over Ω. Evidently P ( . | B) is concentrated on B, i.e. B is a one set with respect to P ( . | B). On the other hand Ω \ B is a null set.

4.10 Exercise S Show that P (A ∪ B) + P (A ∩ B) = P (A) + P (B).