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the sample variance, is an ancillary statistic – its distribution does not depend on μ. Therefore, from Basu's theorem it follows that these statistics are independent conditional on , conditional on . This independence result can also be proven by Cochran's theorem.
In mathematics, the following inequality is known as Titu's lemma, Bergström's inequality, Engel's form or Sedrakyan's inequality, respectively, referring to the article About the applications of one useful inequality of Nairi Sedrakyan published in 1997, [1] to the book Problem-solving strategies of Arthur Engel published in 1998 and to the book Mathematical Olympiad Treasures of Titu ...
In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem " or an "auxiliary theorem".
In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt 's eponymous identity . Notation
The lemma is also valid for the stratification that satisfies Bekka's condition (C), which is weaker than Whitney's condition (B). [5] (The significance of this is that the consequences of the first isotopy lemma cannot imply Whitney’s condition (B).) Thom's second isotopy lemma is a family version of the first isotopy lemma.
An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting = (), the group of invertible 2 × 2 complex matrices, = (), the subgroup of determinant 1 matrices, and the normal subgroup of scalar matrices = {():}, we have = {}, where is ...
In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space.
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function. [ 1 ] Urysohn's lemma is commonly used to construct continuous functions with various properties on normal spaces.