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Without loss of generality, let U = {1, 2, . . , n}, and π : U → [0, 1]. , for A ⊆ 2U , Pf (A) = f (A). , f is a probability density on 2U having precisely π as its covering function. Such an f is determined completely from the joint distribution of the Bernoulli random vector (I1 , I2 , . . , In ). This can be seen as follows. Exhibiting the following bijection between 2U and {0, 1}n : for A ⊆ U , we associate ε = (ε1 , ε2 , . . , εn ) ∈ {0, 1}n where εi = 1 or 0 according to i ∈ A © 2006 by Taylor & Francis Group, LLC 42 An Introduction to Random Sets or not; and conversely, for ε = (ε1 , ε2 , .

In its simplest form, a decision problem consists of choosing an action among a collection of relevant actions Γ in such a way that utility is maximized. Specifically, if Θ denotes the collection of possible “states of nature,” the true value θ0 being unknown, then a utility function u:Γ×Θ→R is specified, where u(a, θ) is a “payoff” when action a is chosen and nature presents θ. In the Bayesian framework, the knowledge about Θ is described by a probability P on Θ. Then the expected value EP u(a, ·) of the utility function u(a, θ) is used to make a choice as to which action a to take.

N. Then   n f (A) = F (A)− i=1 (noting that b) Now n n i=1 i

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50 Jahre Deutsche Statistische Gesellschaft Tagungen 1961 und 1962

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