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The sign of the covariance of two random variables X and Y. In probability theory and statistics, covariance is a measure of the joint variability of two random variables. [1] The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables.
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
[a] This means that the function that maps y to f(x) + J(x) ⋅ (y – x) is the best linear approximation of f(y) for all points y close to x. The linear map h → J(x) ⋅ h is known as the derivative or the differential of f at x. When m = n, the Jacobian matrix is square, so its determinant is a well-defined function of x, known as the ...
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
That is, for any two random variables X 1, X 2, both have the same probability distribution if and only if =. [citation needed] If a random variable X has moments up to k-th order, then the characteristic function φ X is k times continuously differentiable on the entire real line.
For any smooth function f on a Riemannian manifold (M, g), the gradient of f is the vector field ∇f such that for any vector field X, (,) =, that is, ((),) = (), where g x ( , ) denotes the inner product of tangent vectors at x defined by the metric g and ∂ X f is the function that takes any point x ∈ M to the directional derivative of f ...
Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs; Continuous linear operator; Densely defined operator – Function that is defined almost everywhere (mathematics) Hahn–Banach theorem – Theorem on extension of bounded linear functionals
The features of the graph = = + can be interpreted in terms of the variables x and y. The y -intercept is the initial value y = f ( 0 ) = b {\displaystyle y=f(0)=b} at x = 0 {\displaystyle x=0} . The slope a measures the rate of change of the output y per unit change in the input x .