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Given a random variable X ~ Norm[μ,σ] (a normal distribution with mean μ and standard deviation σ) and a constant L > μ, it can be shown via integration by substitution: [] = + (()) where A and B are certain numeric constants.
In calculus, the derivative of any linear combination of functions equals the same linear combination of the derivatives of the functions; [1] this property is known as linearity of differentiation, the rule of linearity, [2] or the superposition rule for differentiation. [3]
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function.
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.
Linear regression can be used to estimate the values of β 1 and β 2 from the measured data. This model is non-linear in the time variable, but it is linear in the parameters β 1 and β 2; if we take regressors x i = (x i1, x i2) = (t i, t i 2), the model takes on the standard form
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if is a holomorphic function, real-valued on the real line, which can be evaluated at points in the complex plane near , then there are stable methods.
The intuition of the delta method is that any such g function, in a "small enough" range of the function, can be approximated via a first order Taylor series (which is basically a linear function). If the random variable is roughly normal then a linear transformation of it is also normal. Small range can be achieved when approximating the ...
Let be a Banach space, let ′ be the dual space of , let : ′ be a linear map, and let ′.A vector is a solution of the equation = if and only if for all , () = ().A particular choice of is called a test vector (in general) or a test function (if is a function space).