Whittle, P., Probability via Expectation, SpringerVerlag, NY, NY, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is close to the expectation. The previous proposition shows that the support can be defined as either a probability (of the supporting set) or an expectation (of the predicate) In fact, probability theory itself can be either based on the concept of probability or the concept of expectation [72]. A second class of examples is obtained when we consider a sample of independant observations as our objects. Thus the observed feature is the sequence x . We consider the relative frequency of a on the observations, i.e. s = as function defined on this object. The probability ....

Peter Whittle. Probability via expectation. Springer Texts in Statistics. Springer-Verlag, New York, fourth edition, 2000.

....on M such that the sets consisting of all 2 M with x( resp. x( are measurable, and x 2 x i Pr(x 2 A) A) for all measurable subsets A of M i Z d ( f( 31) for all bounded measurable real valued functions f on M . Here hf(x)i denotes the expectation of f(x) See Whittle [2] for an introduction to probability theory by means of expectations. Proof. For any interval I [0; 1] we de ne Xj I : f 2 M j x( Ig: 32) Let X be the set of regular points of x. Regularity implies Pr(x(x) Pr(x(x) if 2 X ; 33) and from this we can deduce for ....

P. Whittle, Probability via Expectation, Springer Texts in Statistics, Springer, Berlin 1992.

.... problems, for which more and more ecient algorithms become available [1, 6, 16, 29] A cloud is a new, easily visualized concept for uncertainty with a well de ned semantics, mediating between the concept of a fuzzy set (see, e.g. 9, 18] and that of a probability distribution (see, e.g. [34]) In general, it contains more information than a fuzzy set but less than a distribution. This is in contrast with fuzzy intervals [18, p.16] which, although having the same formal de nition, have a completely di erent interpretation that contains even less information than a fuzzy set. A ....

P. Whittle, Probability via Expectation, Springer Texts in Statistics, Springer, Berlin 1992.

....states T S we pun T with its characteristic function in ES, thus writing T:i = 1 or 0 depending on whether i is a member of T or not. More generally than (1) we invariably write f:x for function f applied to argument x, whatever the context. We use : for is defined to be . It is well known [13] that there is an exact correspondence between ordinary Markov processes and expectation operators t satisfying scaling and constant distribution as set out in Fig. 1, together with the linearity condition t: A B) j t:A t:B ; scaling t: cA) j c(t:A) constant distribution t: A c) j ....

P. Whittle. Probability via expectations. Wiley, second edition, 1980.

....q. 6 Now evaluating at b = false we deduce from the above equality that q = 1 q=2 ; giving q = 2, and (unsurprisingly) an upper bound of 2 on the number of executions of Chooser required to achieve success. 6 Scaling is a standard property of expectation operators from probability theory [18] which also holds here. Others are monotonicity and continuity. In fact only distribution of addition fails: nondeterminism forces a weakening of that axiom; compare suplinearity of Fig. 7. 1: if T i l i 1=2 Phi r i 2: l i r i ) if (l i :R i Gamma1 ) L i [ l i R i Gamma1 ) l ....

P. Whittle. Probability via expectations. Wiley, second edition, 1980.

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Whittle, P., Probability via Expectation, SpringerVerlag, NY, NY, 1992.

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P. Whittle, Probability via expectation, 3rd ed., Springer, New York 1992.

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Peter Whittle. Probability via expectation. Springer Texts in Statistics. Springer-Verlag, New York, fourth edition, 2000.

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