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The proposition in probability theory known as the law of total expectation, [1] the law of iterated expectations [2] (LIE), Adam's law, [3] the tower rule, [4] and the smoothing theorem, [5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then
Many mathematicians then attempted to construct elementary proofs of the theorem, without success. G. H. Hardy expressed strong reservations; he considered that the essential "depth" of the result ruled out elementary proofs: No elementary proof of the prime number theorem is known, and one may ask whether it is reasonable to expect one.
For many purposes, it is only necessary to know that an expansion for in terms of iterated commutators of and exists; the exact coefficients are often irrelevant. (See, for example, the discussion of the relationship between Lie group and Lie algebra homomorphisms in Section 5.2 of Hall's book, [2] where the precise coefficients play no role in the argument.)
The Principles and Standards for School Mathematics was developed by the NCTM. The NCTM's stated intent was to improve mathematics education. The contents were based on surveys of existing curriculum materials, curricula and policies from many countries, educational research publications, and government agencies such as the U.S. National Science Foundation. [3]
The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables. [ citation needed ] One author uses the terminology of the "Rule of Average Conditional Probabilities", [ 4 ] while another refers to it as the "continuous law of ...
Elementary recursive arithmetic (ERA) is a subsystem of primitive recursive arithmetic (PRA) in which recursion is restricted to bounded sums and products. This also has the same Π 2 0 {\displaystyle \Pi _{2}^{0}} sentences as EFA, in the sense that whenever EFA proves ∀x∃y P ( x , y ), with P quantifier-free, ERA proves the open formula P ...
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Where "" is a metalogical symbol representing "can be replaced in a proof with." In strict terminology, ( ( P ∧ Q ) → R ) ⇒ ( P → ( Q → R ) ) {\displaystyle ((P\land Q)\to R)\Rightarrow (P\to (Q\to R))} is the law of exportation, for it "exports" a proposition from the antecedent of ( P ∧ Q ) → R {\displaystyle (P\land Q)\to R} to ...