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In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence (m 0, m 1, m 2, ...), does there exist a positive Borel measure μ (for instance, the measure determined by the cumulative distribution function of a random variable) on the real line such that
Though there are many approximate solutions (such as Welch's t-test), the problem continues to attract attention [4] as one of the classic problems in statistics. Multiple comparisons: There are various ways to adjust p-values to compensate for the simultaneous or sequential testing of hypotheses. Of particular interest is how to simultaneously ...
Central limit theorem; Characterization of probability distributions; Cochran's theorem; Complete class theorem; Continuous mapping theorem; Cox's theorem; Cramér's decomposition theorem; Craps principle
Any classical solution is a mild solution. A mild solution is a classical solution if and only if it is continuously differentiable. [6] The following theorem connects abstract Cauchy problems and strongly continuous semigroups. Theorem: [7] Let A be a closed operator on a Banach space X. The following assertions are equivalent:
The second HK theorem defines an energy functional for the system and proves that the ground-state electron density minimizes this energy functional. In work that later won them the Nobel prize in chemistry , the HK theorem was further developed by Walter Kohn and Lu Jeu Sham to produce Kohn–Sham DFT (KS DFT).
The 6th problem concerns the axiomatization of physics, a goal that 20th-century developments seem to render both more remote and less important than in Hilbert's time. Also, the 4th problem concerns the foundations of geometry, in a manner that is now generally judged to be too vague to enable a definitive answer.
The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. [1] [2] The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that satisfy the basic principles stated for these different approaches.
In statistics, the reference class problem is the problem of deciding what class to use when calculating the probability applicable to a particular case.. For example, to estimate the probability of an aircraft crashing, we could refer to the frequency of crashes among various different sets of aircraft: all aircraft, this make of aircraft, aircraft flown by this company in the last ten years ...