A comparison of Figs. 8 and 9 shows that the interval of σ having the optimal index function values is contained in the interval of σ having the highest clustering accuracy, which illustrates that it is reasonable to use the index function as the fitness function to select the proper σ.
In contrast to this, we prove that the H∞ calculus is bounded over any class of functions whose Fourier spectrum is contained in some interval with 0<ε<σ<∞.
Also, the probability of selecting exactly the number x must be 0, because the set consisting of x alone is contained in the interval [x, x + 1/n] for all n and hence must have no larger probability than 1/[n(s − r)], no matter how large n is.
Through the paper, we let be a standard mollifier sequence in, such that the support of is contained in the interval.
For example, the lower limit θ LL and upper limit θ UL of a 95% symmetrical credible interval are defined by F Θ (θ LL | x) = 0.025 and F Θ (θ UL | x) = 0.975; the probability, given the data, that θ is contained in this interval is 95%.
In fact, it is contained in the interval [m,M] where :m=\inf_{\|x\|=1}\langle Tx, x\rangle,\quad M=\sup_{\|x\|=1}\langle Tx, x\rangle.
For example, the subset of the integers {0, 1, 2} is contained in the interval of real numbers [0, 2], which is convex.
For example, a solid disk \bullet is convex but a circle \circ is not, because it does not contain a line segment joining its points \oslash; the non-convex set of three integers {0, 1, 2} is contained in the interval [0, 2], which is convex.
Since 27.98 is contained in the interval (−55.45, 35.27), the WSRT evaluation of significance is that the Anoka County rate was not unusual.
This criterion requires that the error rates are controlled for any level α of significance, if the empirical rate α ^ is contained in the interval 0.5 α ≤ α ^ ≤ 1.5 α.
Therefore, the probability that the real mean is contained in the confidence interval is P − z 1 − α 2 ≤ μ ^ − μ σ ^ 2 N ≤ z 1 − α 2 ≈ 1 − α (1).
