The Normal (or Gaussian) Distribution and Other Continuous Distributions

2001 ◽  
pp. 44-57
Author(s):  
Byron P. Roe
Keyword(s):  
Gaussian Distribution ◽  
Continuous Distributions

Structure images of Si, Ge and Mos2 crystals and some application to radiation damage studies

1978 ◽  
Vol 36 (1) ◽  
pp. 292-293 ◽  
Author(s):  
K. Izui ◽  
T. Nishida ◽  
S. Furuno ◽  
H. Otsu ◽  
S. Kuwabara
Keyword(s):  
Radiation Damage ◽  
Exposure Time ◽  
Gaussian Distribution ◽  
Intensity Distribution ◽  
Chromatic Aberration ◽  
Magnetic Lens

Recently we have observed the structure images of silicon in the (110), (111) and (100) projection respectively, and then examined the optimum defocus and thickness ranges for the formation of such images on the basis of calculations of image contrasts using the n-slice theory. The present paper reports the effects of a chromatic aberration and a slight misorientation on the images, and also presents some applications of structure images of Si, Ge and MoS2 to the radiation damage studies.(1) Effect of a chromatic aberration and slight misorientation: There is an inevitable fluctuation in the amount of defocus due to a chromatic aberration originating from the fluctuations both in the energies of electrons and in the magnetic lens current. The actual image is a results of superposition of those fluctuated images during the exposure time. Assuming the Gaussian distribution for defocus, Δf around the optimum defocus value Δf0, the intensity distribution, I(x,y) in the image formed by this fluctuation is given by

Limit tolerances for sums of measurement errors

2018 ◽  
Vol 934 (4) ◽  
pp. 59-62
Author(s):  
V.I. Salnikov
Keyword(s):  
Normal Distribution ◽  
Mean Square Error ◽  
Gaussian Distribution ◽  
Measurement Errors ◽  
Theory And Practice ◽  
Mean Square ◽  
The Mean ◽  
Limiting Values ◽  
Limiting Value

The question of calculating the limiting values of residuals in geodesic constructions is considered in the case when the limiting value for measurement errors is assumed equal to 3m, ie ∆рred = 3m, where m is the mean square error of the measurement. Larger errors are rejected. At present, the limiting value for the residual is calculated by the formula 3m√n, where n is the number of measurements. The article draws attention to two contradictions between theory and practice arising from the use of this formula. First, the formula is derived from the classical law of the normal Gaussian distribution, and it is applied to the truncated law of the normal distribution. And, secondly, as shown in [1], when ∆рred = 2m, the sums of errors naturally take the value equal to ?pred, after which the number of errors in the sum starts anew. This article establishes its validity for ∆рred = 3m. A table of comparative values of the tolerances valid and recommended for more stringent ones is given. The article gives a graph of applied and recommended tolerances for ∆рred = 3m.

scholarly journals Separate Universe calibration of the dependence of halo bias on cosmic web anisotropy

2020 ◽  
Vol 499 (3) ◽  
pp. 4418-4431 ◽  
Author(s):  
Sujatha Ramakrishnan ◽  
Aseem Paranjape
Keyword(s):  
Dark Matter ◽  
High Precision ◽  
Gaussian Distribution ◽  
Initial Perturbation ◽  
Analytical Framework ◽  
Web Environment ◽  
Wide Range ◽  
Galaxy Assembly ◽  
Very High ◽  
Good Agreement

ABSTRACT We use the Separate Universe technique to calibrate the dependence of linear and quadratic halo bias b1 and b2 on the local cosmic web environment of dark matter haloes. We do this by measuring the response of halo abundances at fixed mass and cosmic web tidal anisotropy α to an infinite wavelength initial perturbation. We augment our measurements with an analytical framework developed in earlier work that exploits the near-lognormal shape of the distribution of α and results in very high precision calibrations. We present convenient fitting functions for the dependence of b1 and b2 on α over a wide range of halo mass for redshifts 0 ≤ z ≤ 1. Our calibration of b2(α) is the first demonstration to date of the dependence of non-linear bias on the local web environment. Motivated by previous results that showed that α is the primary indicator of halo assembly bias for a number of halo properties beyond halo mass, we then extend our analytical framework to accommodate the dependence of b1 and b2 on any such secondary property that has, or can be monotonically transformed to have, a Gaussian distribution. We demonstrate this technique for the specific case of halo concentration, finding good agreement with previous results. Our calibrations will be useful for a variety of halo model analyses focusing on galaxy assembly bias, as well as analytical forecasts of the potential for using α as a segregating variable in multitracer analyses.

Dispersive distributions, and the connection between dispersivity and strong unimodality

1981 ◽  
Vol 18 (01) ◽  
pp. 76-90 ◽  
Author(s):  
Toby Lewis ◽  
J. W. Thompson
Keyword(s):  
Continuous Distributions ◽  
Strong Unimodality

Two continuous distributions, G, H so related that any two quantiles of H are more widely separated than the corresponding quantiles of G may be said to be ‘ordered in dispersion'; Saunders and Moran have given examples. It is shown here that distributions F (called ‘dispersive' distributions) exist, e.g. the exponential, such that if G, H are ordered in dispersion then so also are the convolutions F ∗G, F ∗H. The class of dispersive distributions is determined, and shown to coincide with the class of strongly unimodal distributions.

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