introduction-probability.pdf

1 a map p f 0 1 is called probability measure if p ω

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(1) A map P : F → [0 , 1] is called probability measure if P (Ω) = 1 and for all A 1 , A 2 , ... ∈ F with A i A j = for i = j one has P i =1 A i = i =1 P ( A i ) . (1.1) The triplet (Ω , F , P ) is called probability space . (2) A map μ : F → [0 , ] is called measure if μ ( ) = 0 and for all A 1 , A 2 , ... ∈ F with A i A j = for i = j one has μ i =1 A i = i =1 μ ( A i ) . The triplet (Ω , F , μ ) is called measure space . (3) A measure space (Ω , F , μ ) or a measure μ is called σ -finite provided that there are Ω k Ω, k = 1 , 2 , ... , such that (a) Ω k ∈ F for all k = 1 , 2 , ... , (b) Ω i Ω j = for i = j , (c) Ω = k =1 Ω k , (d) μ k ) < . The measure space (Ω , F , μ ) or the measure μ are called finite if μ (Ω) < . Remark 1.2.2 Of course, any probability measure is a finite measure: We need only to check μ ( ) = 0 which follows from μ ( ) = i =1 μ ( ) (note that ∅ ∩ ∅ = ) and μ (Ω) < . Example 1.2.3 ( a ) We assume the model of a die, i.e. Ω = { 1 , ..., 6 } and F = 2 Ω . Assuming that all outcomes for rolling a die are equally likely, leads to P ( { ω } ) := 1 6 . Then, for example, P ( { 2 , 4 , 6 } ) = 1 2 .
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1.2. PROBABILITY MEASURES 15 ( b ) If we assume to have two coins, i.e. Ω = { ( T, T ) , ( H, T ) , ( T, H ) , ( H, H ) } and F = 2 Ω , then the intuitive assumption ’fair’ leads to P ( { ω } ) := 1 4 . That means, for example, the probability that exactly one of two coins shows heads is P ( { ( H, T ) , ( T, H ) } ) = 1 2 . Example 1.2.4 [Dirac and counting measure] 2 ( a ) Dirac measure : For F = 2 Ω and a fixed x 0 Ω we let δ x 0 ( A ) := 1 : x 0 A 0 : x 0 A . ( b ) Counting measure : Let Ω := { ω 1 , ..., ω N } and F = 2 Ω . Then μ ( A ) := cardinality of A. Let us now discuss a typical example in which the σ –algebra F is not the set of all subsets of Ω. Example 1.2.5 Assume that there are n communication channels between the points A and B . Each of the channels has a communication rate of ρ > 0 (say ρ bits per second), which yields to the communication rate ρk , in case k channels are used. Each of the channels fails with probability p , so that we have a random communication rate R ∈ { 0 , ρ, ..., nρ } . What is the right model for this? We use Ω := { ω = ( ε 1 , ..., ε n ) : ε i ∈ { 0 , 1 } ) with the interpretation: ε i = 0 if channel i is failing, ε i = 1 if channel i is working. F consists of all unions of A k := { ω Ω : ε 1 + · · · + ε n = k } . Hence A k consists of all ω such that the communication rate is ρk . The system F is the system of observable sets of events since one can only observe how many channels are failing, but not which channel fails. The measure P is given by P ( A k ) := n k p n - k (1 - p ) k , 0 < p < 1 . Note that P describes the binomial distribution with parameter p on { 0 , ..., n } if we identify A k with the natural number k . 2 Paul Adrien Maurice Dirac, 08/08/1902 (Bristol, England) - 20/10/1984 (Tallahassee, Florida, USA), Nobelprice in Physics 1933.
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16 CHAPTER 1. PROBABILITY SPACES We continue with some basic properties of a probability measure. Proposition 1.2.6 Let , F , P ) be a probability space. Then the following assertions are true: (1) If A 1 , ..., A n ∈ F such that A i A j = if i = j , then P ( n i =1 A i ) = n i =1 P ( A i ) .
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