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[8] [9] [verification needed] Cramer's rule can also be numerically unstable even for 2×2 systems. [10] However, Cramer's rule can be implemented with the same complexity as Gaussian elimination, [11] [12] (consistently requires twice as many arithmetic operations and has the same numerical stability when the same permutation matrices are ...
The number of distinct terms (including those with a zero coefficient) in an n-th degree equation in two variables is (n + 1)(n + 2) / 2.This is because the n-th degree terms are ,, …,, numbering n + 1 in total; the (n − 1) degree terms are ,, …,, numbering n in total; and so on through the first degree terms and , numbering 2 in total, and the single zero degree term (the constant).
Cramér’s decomposition theorem, a statement about the sum of normal distributed random variable; Cramér's theorem (large deviations), a fundamental result in the theory of large deviations; Cramer's theorem (algebraic curves), a result regarding the necessary number of points to determine a curve
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, [1] is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that
The logarithmic moment generating function (which is the cumulant-generating function) of a random variable is defined as: = [ ()].Let ,, … be a sequence of iid real random variables with finite logarithmic moment generating function, i.e. () < for all .
It turns out that the converse is also true. The latter result, initially announced by Paul Lévy, [1] has been proved by Harald Cramér. [2] This became a starting point for a new subfield in probability theory, decomposition theory for random variables as sums of independent variables (also known as arithmetic of probabilistic distributions). [3]
The equations 3x + 2y = 6 and 3x + 2y = 12 are inconsistent. A linear system is inconsistent if it has no solution, and otherwise, it is said to be consistent. [7] When the system is inconsistent, it is possible to derive a contradiction from the equations, that may always be rewritten as the statement 0 = 1. For example, the equations
The result is named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, [1] [2] [3] but has also been derived independently by Maurice Fréchet, [4] Georges Darmois, [5] and by Alexander Aitken and Harold Silverstone. [6] [7] It is also known as Fréchet-Cramér–Rao