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However, a finitely additive set function might not have the additivity property for a union of an infinite number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, μ ( ⋃ n = 1 ∞ A n ) = ∑ n = 1 ∞ μ ( A n ) . {\textstyle \mu \left(\bigcup _{n=1}^{\infty ...
However if we held a portfolio that consisted of 50% of each bond by value then the 95% VaR is 35% (= 0.5*0.7 + 0.5*0) since the probability of at least one of the bonds defaulting is 7.84% (= 1 - 0.96*0.96) which exceeds 5%. This violates the sub-additivity property showing that VaR is not a coherent risk measure.
Intuitively, the additivity property says that the probability assigned to the union of two disjoint (mutually exclusive) events by the measure should be the sum of the probabilities of the events; for example, the value assigned to the outcome "1 or 2" in a throw of a dice should be the sum of the values assigned to the outcomes "1" and "2".
Proof of countable additivity. One automatically has the conclusion in the form " ≤ {\displaystyle \,\leq \,} " from the definition of outer measure. So it is only necessary to prove the " ≥ {\displaystyle \,\geq \,} " inequality.
Coombs' theory in all three studies was applied to a set of six statements. These authors found that the axioms were satisfied, however, these were applications biased towards a positive result. With six stimuli, the probability of an interstimulus midpoint order satisfying the double cancellation axioms at random is .5874 (Michell, 1994).
In mathematics, in particular in measure theory, a content is a real-valued function defined on a collection of subsets such that [,].() =(=) = = (), …,, = =.That is, a content is a generalization of a measure: while the latter must be countably additive, the former must only be finitely additive.
As mentioned above, additivity is a property of cardinal utility functions. An analogous property of ordinal utility functions is weakly additive . A utility function is additive if and only if it is both submodular and supermodular .
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.