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The value of () can be given by several series. In terms of a sum involving the floor function it can be expressed as: [5] = + = (⌊ + ⌋ ⌊ + ⌋).This is a consequence of Jacobi's two-square theorem, which follows almost immediately from the Jacobi triple product.
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 ...
The Normal PDF does not describe this derived data particularly well, especially at the low end. Substituting the known mean (10) and variance (4) of the x values in this simulation, or in the expressions above, it is seen that the approximate (1600) and exact (1632) variances only differ slightly (2%).
The area contained by both circles is the joint entropy H(X,Y). The circle on the left (red and violet) is the individual entropy H(X), with the red being the conditional entropy H(X|Y). The circle on the right (blue and violet) is H(Y), with the blue being H(Y|X). The violet is the mutual information I(X;Y). In information theory, joint ...
We will write X ~ B to mean that X is a random variable whose distribution function is unknown except that it is inside B. Thus, X ~ F ∈ B can be contracted to X ~ B without mentioning the distribution function explicitly. If X and Y are independent random variables with distributions F and G respectively, then X + Y = Z ~ H given by
Given some experimental measurements of a system and some computer simulation results from its mathematical model, inverse uncertainty quantification estimates the discrepancy between the experiment and the mathematical model (which is called bias correction), and estimates the values of unknown parameters in the model if there are any (which ...
The relation between and are given by the following table, where the values for DRMS and 2DRMS (twice the distance root mean square) are specific to the Rayleigh distribution and are found numerically, while the CEP, R95 (95% radius) and R99.7 (99.7% radius) values are defined based on the 68–95–99.7 rule
The smallest-circle problem (also known as minimum covering circle problem, bounding circle problem, least bounding circle problem, smallest enclosing circle problem) is a computational geometry problem of computing the smallest circle that contains all of a given set of points in the Euclidean plane.