Let X and Y be random variables (discrete or continuous!) with means μ X and μ Y. The covariance of X and Y, denoted Cov ( X, Y) or σ X Y, is defined as: C o v ( X, Y) = σ X Y = E [ ( X − μ X) ( Y − μ Y)] That is, if X and Y are discrete random variables with joint support S, then the covariance of X and Y is: C o v ( X, Y) = ∑ ∑ ( x, y) ∈ S. .
Aug 28, 2017 · Explanation: Constants are numbers by themselves. The constant in this expression is: 60. Variables are letters or symbols (not numbers) in an algebraic expression. The variables in the expression are: x and y. Coefficients are number paired with variables. The variables in this expression are: −6 and −1.
Steps for Solving Linear Equation. x+y=xy. x + y = x y. Subtract xy from both sides. Subtract x y from both sides. x+y-xy=0. x + y − x y = 0. Subtract x from both sides. Anything subtracted from zero gives its negation.
Example 5.6.1 A Linear Equation with Variable Coefficients. Suppose we're given the IVP y ″ + t2y = 0; y(0) = A, y ′ (0) = B. First of all, note that this seems to be a fairly simple linear equation with variable coefficients. There is only one variable coefficient, a simple polynomial, and the forcing function is zero.
During the past years, multiscale methods have been proven to be asymptotically optimal efficient numerical schemes for elliptic partial differential equations. This means that the resulting linear systems of equations can be solved with an overall amount of work which is of the order of the number of unknowns. In particular, this implies that the number of iterations for an iterative scheme is independent of the scale. Discretizations allowing such a multilevel structure include uniformly refined finite elements and bases of multiresolution spaces. The latter may be used as a nested sequence of trial spaces where bases, called wavelets, of the complement between two succeeding spaces are available. This availability of direct sum decompositions together with additional analytic properties provide a powerful tool to prove the theoretical asymptotical optimality of the resulting numerical schemes for a whole range of operator equations, including elliptic partial differential and singular integral equations.
Although the Laplace transform can be used to solve certain linear differential equations with variable coefficients, the calculations may be very complicated and ultimately frustrating. As a general rule, Laplace transform techniques are not useful for most problems of this type.