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In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. [1][2] For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously ...
Cramer's rule is a closed-form expression, in terms of determinants, of the solution of a system of n linear equations in n unknowns. Cramer's rule is useful for reasoning about the solution, but, except for n = 2 or 3 , it is rarely used for computing a solution, since Gaussian elimination is a faster algorithm.
Cramér–Rao bound. Illustration of the Cramer-Rao bound: there is no unbiased estimator which is able to estimate the (2-dimensional) parameter with less variance than the Cramer-Rao bound, illustrated as standard deviation ellipse. In estimation theory and statistics, the Cramér–Rao bound (CRB) relates to estimation of a deterministic ...
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that ...
Ruin theory. In actuarial science and applied probability, ruin theory (sometimes risk theory[1] or collective risk theory) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at ...
Cramér's theorem (large deviations) Cramér's theorem is a fundamental result in the theory of large deviations, a subdiscipline of probability theory. It determines the rate function of a series of iid random variables. A weak version of this result was first shown by Harald Cramér in 1938.
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj (A), is the transpose of its cofactor matrix. [1][2] It is occasionally known as adjunct matrix, [3][4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.