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Topic: Unimodal distribution

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More thoughts on a one tailed version of Chebyshev's inequality - by Henry Bottomley What follows is largely essentially a graphical proof, comparing a general continuous random variable with a monotonically decreasing or unimodal probability density function which satisfies the given conditions against either a uniform distribution or a uniform distribution with a point of positive probability at one end. The key initial step is to show that for any unimodal distribution, there is another (consisting of a point of positive probability combined with a uniform distribution of the rest of the probability which has the same value for P[X-E(X)>=t] and an equal or lower variance. <=3.Var(X) implying that for a continuous unimodal distribution: www.btinternet.com /~se16/hgb/cheb2.htm

More mean median mode relationships - by Henry Bottomley It is widely believed that the median of a unimodal distribution is "usually" between the mean and the mode. I earlier produced a page on Chebyshev's inequality in both its original two-tailed version and a one-tailed version, another page on the difference between the mean and mode in a unimodal distribution, another on Chebyshev type inequalities for unimodal distributions. This page considers how the mean, median, mode, and standard deviation affect each other in a unimodal distribution, and puts a limit on the median when the mode equals the mean. www.btinternet.com /~se16/hgb/median.htm

More thoughts on a one tailed version of Chebyshev's inequality - by Henry Bottomley What follows is largely essentially a graphical proof, comparing a general continuous random variable with a monotonically decreasing or unimodal probability density function which satisfies the given conditions against either a uniform distribution or a uniform distribution with a point of positive probability at one end. The key initial step is to show that for any unimodal distribution, there is another (consisting of a point of positive probability combined with a uniform distribution of the rest of the probability which has the same value for P[X-E(X)>=t] and an equal or lower variance. What this says is that for a given mean, we minimise the second moment by having all the probability as close to 0 as possible, which happens with a uniform distribution. www.btinternet.com /~se16/hgb/cheb2.htm

Glossary for ordination terms Also, species with unimodal distributions may appear to have monotonic distributions may appear to have short gradients if only a small portion of the gradient is sampled. It is caused by the unimodal distribution of species along gradients. Some ordination techniques (such as DCA and CCA) perform best when species have unimodal distributions, others (such as PCA and RDA) perform better when species have monotonic distributions along gradients (i.e. www.okstate.edu /artsci/botany/ordinate/glossary.htm

Publications List While this leads to a tractable algorithm, the mean field distribution is assumed to be factorial and hence unimodal. In this paper we demonstrate the feasibility of using a richer class of approximating distributions based on mixtures of mean field distributions. One approach which has been adopted is to bound the log likelihood using a mean-field approximating distribution. www.research.microsoft.com /~cmbishop/publications_abs.htm

Difference between mode and mean - by Henry Bottomley This page puts an upper bound on the difference between the mode and mean of a random variable with a weakly unimodal probability density function or with a weakly unimodal probability distribution, and proves this bound. If this is not seen as being sufficiently unimodal, then equality is not achieved for a continuous random variable except in the trivial case where the distribution is a single atom, since equality requires a=0 in Step 3, and requires f(x)=u(x), possibly except at x=0 and x=2.m/(1-a), in Step 2 For this to be unimodal, the point of positive probability needs to be the mode of the random variable, and the continuous elements need to be weakly monotonically increasing up to that point and weakly monotonically decreasing down from that point. www.btinternet.com /~se16/hgb/mode.htm

glosu.html A typical example is the normal distribution which happens to be also symmetrical but many unimodal distributions are not symmetrical (e.g., typically the distribution of income is not symmetrical but "left-skewed"; see skewness). The input value is passed through the unit's activation function to produce a single output value, also known as the activation level of the unit. The outputs of the units in the preceding layer, the weights on the associated connections, and the threshold value are fed through the unit's synaptic function (post synaptic potential function) to produce a single value (the unit's input value). statsoftinc.com /textbook/glosu.html

Ciencia Abierta Nº 7 The objective of the present work is the determination of the flaw size distribution function in brittle materials subjected to a uniform stress field without postulating previously a known analytical form for the specific risk function of Weibull. Using the integral equations method the flaw size distribution function was determined in the case of brittle materials. This kind of problem implies the use of functions of risk more general than the ones of two and three parameters commonly employed and it can be dealt with by the integral equations method, where the solution function can be expressed in terms of a differential operator applied over a function of risk totally general. encuesta.ing.uchile.cl /~cabierta/revista/7/flaw.htm

Basic Statistics Review - Unit 1 - 112 Asymmetric distributions may also be unimodal, bimodal, or multimodal. A symmetric distribution is one in which the shape of left side of the distribution is a "mirror image" of the right side. A negatively skewed distribution is one in which the left (negative) tail of the distribution is the long one (see Figure 6.2). www.msu.edu /user/sw/statrev/strv112.htm

EPDIRM Manual This is a symmetric unimodal # distribution in the interval 0.0 to 0.5 with mode at 0.25. The default item parameter prior distributions for the a-, b- and c-parameters are four-parameter beta with parameters (1.75, 3.0, 0.0, 3.0), (1.01, 1.01, -6.0, 6.0) and (3.5, 4.0, 0.0, 0.5), respectively, where the four numbers in the parameter vector are the two shape parameters followed by the lower and upper limits of the distribution. For the normal and lognormal distribution the parameters are the mean and variance, for the four-parameter beta distribution the parameters are the two shape parameters, the lower limit, and the upper limit. www.b-a-h.com /software/epdirm/epdirm.html

Ch2 Distributions Pt1 In Figure 2.3, the distributions for Sections A, B, C, D, and E are all unimodal distributions with varying degrees of skew and kurtosis. In a platykurtic distribution the individual measures are spread out fairly uniformly across their range, whereas in a leptokurtic distribution they tend to cluster compactly at some particular point in the range. In the mesokurtic distribution illustrated by Section C the clustering is more moderate than in the leptokurtic distribution, and the curve as it falls away from the peak is more tapering than in the platykurtic distribution. faculty.vassar.edu /lowry/ch2pt1.html

gparetodemo.m For unimodal distributions, such as the normal or % Student's t, these regions are known as the "tails" of the distribution. One approach to distribution fitting % that involves the GPD is to use a non-parametric fit (the empirical % cumulative distribution function, for example) in regions where there are % many observations, and to fit the GPD to the tail(s) of the data. The Generalized Pareto Distribution (GPD) was developed as a % distribution that can model tails of a wide variety of distributions, % based on theoretical arguments. www.clemson.edu /cle4_share/CWE/COES0915_CLUG/REFERENCE/matlabr14/toolbox/stats/gparetodemo.m   (778 words)

Cusp Surface Analysis Thus when the empirical conditional distributions of YX are clearly unimodal there is a great deal of similarity between the effects of A and C. When this occurs, the estimators for the coefficients of these factors may be highly correlated (this occurs primarily in datasets with very few observations in the bimodal zone). In fact, the maximum likelihood method will find the coefficients which best reproduce the empirical conditional distribution of Y, since this is roughly what it means to maximize the likelihood of a model. On the other hand, the likelihood method makes the further assumption that this conditional distribution has a particular form, examples of which are depicted in Figure 4. www.aetheling.com /models/cusp/Intro.htm   (778 words)

The Poisson Distribution Thus, the distribution is unimodal, with the mode occurs at the greatest integer in The Poisson distribution is one of the most important in probability. This result means that, in a sense, the Poisson model gives the most "random" distribution of points in time. www.ds.unifi.it /VL/VL_EN/poisson/poisson4.html   (736 words)

New Page 1 A unimodal distribution is preferred for many statistics. A distribution is considered symmetrical if it’s skew value in SPSS is between -1 and 1. The extent to which cases are piled up around the measure of central tendency or the tails of the distribution. www.sfu.ca /~amalm/320class8.htm   (736 words)

Cauchy distribution The Cauchy distribution is unimodal and symmetric; its distribution function is: www.fundp.ac.be /~fbastin/computing/rangen/doc/node13.html   (27 words)

Citebase - The cycle enumerator of unimodal permutations We give a generating function for the number of unimodal permutations with a given cycle structure. We also obtain in effect a kind of combinatorial universality for continuous unimodal... This paper concerns a probability distribution on the symmetric group generalizing the riffle shuffle of Bayer, Diaconis, and others. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0102051   (27 words)

6. Wavelets -- Chicago This sampling distribution does not depend upon the scale size of the wavelet function. For the particular case of the MH function, we may compute the Fourier Transform analytically; we present the derivation in Appendix 6.6. In appendices which relate to § 6.3, we present derivations of analytic quantities related to the MH function (Appendix 6.6), as well as describe the simulations used to estimate the threshold correlation value for source detection in each pixel (Appendix 6.7). ledas-cxc.star.le.ac.uk /udocs/docs/swdocs/detect/html/node14.html   (27 words)

Nonveridical Visual Perception in Human Amblyopia -- Barrett et al. 44 (4): 1555 -- Investigative Ophthalmology & Visual Science function of orientation, but it is conceivable that the distribution function of the luminance response, as measured with a photometer The ticks on the abscissa of the contrast sensitivity plots correspond to spatial frequencies of 1 and 10 cycles per degree. www.iovs.org /cgi/content/full/44/4/1555   (27 words)

8.1 Monotonic and unimodal smoothing The points indicate the unimodal regression and the solid line a spline smooth of the unimodal regression step function. The universal constants come from the technique used by Mammen, the approximation of the empirical distribution function by a sequence of Brownian bridges. Figure 8.1 shows a spline smooth and a PAV smooth obtained from estimating the Engel curve of food as a function of income. www.quantlet.com /mdstat/scripts/anr/html/anrhtmlnode43.html   (27 words)

Uwe Cantner, Bernd Ebersberger, Horst Hanusch, Jens J. Krüger and Andreas Pyka: Empirically Based Simulation: The Case of Twin Peaks in National Income Drawing on the Summers/Hestons Penn World Table 5.6 ( 1991) we determine kernel density distributions which are able to detect the aforementioned twin peaked structure and show that the world income distribution starting with an unimodal structure in 1960 evolves subsequently to a bimodal or twin-peak structure. Using the so determined transition rates in the synergetic model leads in both cases to the emergence of the bimodal distribution, which, however, is only in the latter case a persistent phenomenon. The distribution of those per capita income levels are shown for the years 1960, 1980 and 1990. jasss.soc.surrey.ac.uk /4/3/9.html   (27 words)

glosu.html A typical example is the normal distribution which happens to be also symmetrical but many unimodal distributions are not symmetrical (e.g., typically the distribution of income is not symmetrical but "left-skewed"; see skewness). The animation above shows the Weibull distribution as the shape parameter increases (.5, 1, 2, 3, 4, 5, and 10). The Wald statistic is tested against the Chi-square distribution. www.statsoft.com /textbook/glosu.html   (2085 words)

RANDOM NUMBER GENERATION (LUC DEVROYE) For many distributions, it is possible to generate a random variate in one assignment statement using only simple mathematical functions and ordinary arithmetic operations. To generate a random vector from a given ortho-unimodal density, several general-purpose algorithms are presented; and an experimental performance evaluation illustrates the potential efficiency increases that can be achieved by these algorithms versus naive rejection. The fixed-points of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. cgm.cs.mcgill.ca /~luc/rng.html   (2085 words)

S-PLUS help The logistic is a unimodal, symmetric distribution on the real line with tails that are longer than the Gaussian distribution. Density, cumulative probability, quantiles and random generation for the logistic distribution. Johnson, N. and Kotz, S. Continuous Univariate Distributions, vol. www.uni-muenster.de /ZIV/Mitarbeiter/BennoSueselbeck/s-html/helpfiles/rlogis.html   (156 words)

glosu.html A typical example is the normal distribution which happens to be also symmetrical but many unimodal distributions are not symmetrical (e.g., typically the distribution of income is not symmetrical but "left-skewed"; see skewness). The Wald statistic is tested against the Chi-square distribution. In several search algorithms, a penalty factor which is multiplied by the number of units in the network and added to the error of the network, when comparing the performance of the network with others. www.statsoft.com /textbook/glosu.html   (156 words)

STAT406 (96F) Solution 4 The distributions are very much alike, The histograms (naive, sequential, exponential spacings from the top) show unimodal distribution centered around two with slight skewness to the right. The histograms and the boxplot reveal that the distributions are more clustered around the center 2 but still slightly skewed to the right. For generating random samples from cauchy distribution, see one claim in Afternote 9. www-personal.umich.edu /~hiroaki/class/96F/stat406/sol/sol4.html   (1325 words)

Cauchy distribution The Cauchy distribution is unimodal and symmetric; its distribution function is: double ran_cauchy(const Random *random, double a, double b) double ran_cauchy_get_parameters(const Random *random, double *a, double *b) www.fundp.ac.be /~fbastin/computing/rangen/doc/node13.html   (1325 words)

Students t test: Like the normal distribution the t distribution is symmetrical, unimodal and bell shaped but with greater spread and different area properties. This follows the t distribution, which are a number of distributions each with a degree of freedom. Tc is the critical t value that cuts 5% from the two tails of the distribution. www.portfolio.mvm.ed.ac.uk /studentwebs/session4/59/students.htm   (225 words)

Yellow sun in blue sky The distribution of ratios for women is unimodal, and broader than that for men. For men, the distribution of ratios is bimodal, falling into two distinct groups, with 60 per cent of the observers in one group and 40 per cent in the other. In the last decade it has been shown that these distributions are correlated with genetically based polymorphisms of longwave and middlewave cone photopigments. www.philosophy.ubc.ca /colour/Indivdif.htm   (225 words)

Coupling of Nonparametric Frequency and L-Moment Analyses for Mixed Distribution Identification However, the nonparametric frequency analysis indicated that a majority of stations followed nonunimodal mixed distributions since peak flows occur during different seasons and are the result of different generating mechanisms. The L-moment analysis concluded that the data were generated from a unimodal Generalized Extreme Value (GEV) distribution. Thus, the nonparametric method helps in identifying underlying probability distribution, especially when samples arise from mixed distributions. www.awra.org /jawra/papers/J91112.html   (158 words)

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