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Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
Total emitted energy, , of a black body as a function of its temperature, . The upper (black) curve depicts the Stefan–Boltzmann law, M ∘ = σ T 4 {\displaystyle M^{\circ }=\sigma \,T^{4}} . The lower (blue) curve is total energy according to the Wien approximation , M W ∘ = M ∘ / ζ ( 4 ) ≈ 0.924 σ T 4 {\displaystyle M_{W}^{\circ ...
MPFR's computation is both efficient and has a well-defined semantics: the functions are completely specified on all the possible operands and the results do not depend on the platform. [3] This is done by copying the ideas from the ANSI/IEEE-754 standard for fixed-precision floating-point arithmetic (correct rounding and exceptions, in ...
Condition numbers can also be defined for nonlinear functions, and can be computed using calculus.The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
Mathematically, a strict power law cannot be a probability distribution, but a distribution that is a truncated power function is possible: () = for > where the exponent (Greek letter alpha, not to be confused with scaling factor used above) is greater than 1 (otherwise the tail has infinite area), the minimum value is needed otherwise the ...
If n is a power of an odd prime number the formula for the totient says its totient can be a power of two only if n is a first power and n − 1 is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these.
The main application of statistical power is "power analysis", a calculation of power usually done before an experiment is conducted using data from pilot studies or a literature review. Power analyses can be used to calculate the minimum sample size required so that one can be reasonably likely to detect an effect of a given size (in other ...
A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density , the normal density , and Student's ...