We discuss properties of two methods for ascribing probabilities to the shape of a probability distribution. One is based on the idea of counting the number of modes of a bootstrap version of a standard kernel density estimator. We argue that the simplest form of that method suffers from the same difficulties that inhibit level accuracy of Silverman’s bandwidth-based test for modality: the cond...

two statistical downscaling models, the non-homogeneous hidden markov model (nhmm) and the statistical down–scaling model (sdsm) were used to generate future scenarios of both mean and extremes in the tarim river basin,which were based on nine combined scenarios including three general circulation models (gcms) (csiro30, echam5,and gfdl21) predictor sets and three special report on emission sce...

High-resolution numerical experiments, described in this work, show that velocity fluctuations governed by the one-dimensional Burgers equation driven by a white-intime random noise with the spectrum |f(k)| ∝ k exhibit a biscaling behavior: All moments of velocity differences Sn≤3(r) = |u(x+ r)− u(x)| n ≡ |∆u| ∝ rn/3, while Sn>3(r) ∝ r ξn with ξn ≈ 1 for real n > 0 (Chekhlov and Yakhot, Phys. R...

In this paper, we present an integrodifferential diffusion equation for continuous-time random walk that is valid for a generic waiting time probability density function. Using this equation, we also study diffusion behaviors for a couple of specific waiting time probability density functions such as exponential and a combination of power law and generalized Mittag-Leffler function. We show tha...

The dynamics of velocity fluctuations governed by the Burgers equation, driven by the white-in-time random forcing function with f fsx 1 r , td 2 fsx, t0dg2 ~ rdst 2 t0d is considered on the interval 0 , x , L. The properties of the probability density function of velocity differences PsDu, rd are investigated for the three cases j ­ h0; 1y2; 2j. It is shown that the tail of the probability den...

The angular turning probability density distribution for motile bacteria is usually measured in local coordinates and is therefore inconvenient for global analyses of the chemotactic bacterial migration. In this paper we present analytical derivations that convert the local angular turning probability density distribution into a global one. The explicit expression of a reduced global turning pr...

Using the idea of the macroscopic quantum wave function and the definition of the classical chaos, we analytically reveal that the probability density of two periodically driven and weakly coupled Bose-Einstein condensates is deterministic but not predictable. Numerical calculation for the time evolutions of the chaotic probability density demonstrates the analytical result and exhibits the non...

cannot be a probability density function. To see this, note 2x− x = −x(x− √ 2)(x+ √ 2). If C = 0, then f(x) = 0 for all x, so ∫∞ −∞ f(x) = 0, but every probability density function has ∫∞ −∞ f(x) = 1. If C > 0, then f(x) < 0 on the range x ∈ (0, √ 2), but every probability density function has f(x) ≥ 0 for all x. If C < 0, then f(x) < 0 on the range x ∈ ( √ 2, 5/2), but every probability densit...