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Convergence of random variables

In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, its applications to statistics and stochastic processes; the same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of random or unpredictable events can sometimes be expected to settle down into a behavior, unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two understood behaviors are that the sequence takes a constant value, that values in the sequence continue to change but can be described by an unchanging probability distribution. "Stochastic convergence" formalizes the idea that a sequence of random or unpredictable events can sometimes be expected to settle into a pattern. The pattern may for instance be Convergence in the classical sense to a fixed value itself coming from a random event An increasing similarity of outcomes to what a purely deterministic function would produce An increasing preference towards a certain outcome An increasing "aversion" against straying far away from a certain outcome That the probability distribution describing the next outcome may grow similar to a certain distributionSome less obvious, more theoretical patterns could be That the series formed by calculating the expected value of the outcome's distance from a particular value may converge to 0 That the variance of the random variable describing the next event grows smaller and smaller.

These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been studied. While the above discussion has related to the convergence of a single series to a limiting value, the notion of the convergence of two series towards each other is important, but this is handled by studying the sequence defined as either the difference or the ratio of the two series. For example, if the average of n independent random variables Yi, i = 1... n, all having the same finite mean and variance, is given by X n = 1 n ∑ i = 1 n Y i as n tends to infinity, Xn converges in probability to the common mean, μ, of the random variables Yi. This result is known as the weak law of large numbers. Other forms of convergence are important in other useful theorems, including the central limit theorem. Throughout the following, we assume, a sequence of random variables, X is a random variable, all of them are defined on the same probability space. With this mode of convergence, we expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution.

Convergence in distribution is the weakest form of convergence discussed, since it is implied by all other types of convergence mentioned in this article. However, convergence in distribution is frequently used in practice. A sequence X1, X2... of real-valued random variables is said to converge in distribution, or converge weakly, or converge in law to a random variable X if lim n → ∞ F n = F, for every number x ∈ R at which F is continuous. Here Fn and F are the cumulative distribution functions of random variables X, respectively; the requirement that only the continuity points of F should be considered is essential. For example, if Xn are distributed uniformly on intervals this sequence converges in distribution to a degenerate random variable X = 0. Indeed, Fn = 0 for all n when x ≤ 0, Fn = 1 for all x ≥ 1/n when n > 0. However, for this limiting random variable F = 1 though Fn = 0 for all n, thus the convergence of cdfs fails at the point x = 0. Convergence in distribution may be denoted as where L X is the law of X.

For example, if X is standard normal we can write X n → d N. For random vectors ⊂ Rk the convergence in distribution is defined similarly. We say that this sequence converges in distribution to a random k-vector X if lim n → ∞ Pr ⁡ = Pr ⁡ for every A ⊂ Rk, a continuity set of X; the definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, to the “random variables” which are not measurable — a situation which occurs for example in the st

Public Service Medal (Australia)

The Public Service Medal is a civil decoration awarded to Australian public servants for outstanding service. The PSM was introduced in 1989 and replaced the Imperial awards discontinued in 1975, supplementing the Order of Australia introduced that same year. Recipients of the Public Service Medal are entitled to use the post-nominal letters "PSM"; the medal is awarded twice each year by the Governor-General of Australia, on the nomination of the responsible Minister in each state or territory, at the federal level. The total number of awards made each year must not exceed 100, further broken down into a quota for each government public service; the Public Service Medal is a circular nickel-silver medal ensigned with a Federation Star. The obverse shows an inner circle with four planetary gears spaced around a sun gear, it is surrounded by the words'Public Service'. An outer circle shows 36 human figures symbolising a range of activities; the reverse displays a wreath of mimosa surrounding the inscription'For Outstanding Service'.

The 32 millimetre-wide ribbon features the national colours of green and gold in a vertical striped pattern. Australian Honours Order of Precedence

Simple interactive object extraction

Simple interactive object extraction is an algorithm for extracting foreground objects from color images and videos with little user interaction. It has been implemented as "foreground selection" tool in the GIMP, as part of the tracer tool in Inkscape, as function in ImageJ and Fiji. Experimental implementations were reported for Blender and Krita. Although the algorithm was designed for videos all implementations use SIOX for still image segmentation. In fact, it is said to be the current de facto standard for this task in the open-source world. A free hand selection tool is used to specify the region of interest, it must contain all foreground objects to extract and as few background as possible. The pixels outside the region of interest form the sure background while the inner region define a superset of the foreground, i.e. the unknown region. A so-called foreground brush is used to mark representative foreground regions; the algorithm outputs a selection mask. The selection can be refined by either adding further foreground markings or by adding background markings using the background brush.

Technically, the algorithm performs the following steps: Create a set of representative colors for sure foreground and sure background, the so-called color signatures. Assign all image points to foreground or background by a weighted nearest neighbor search in the color signatures. Apply some standard image processing operations like erode and blur to remove artifacts. Find the connected foreground components that are either large enough or marked by the user. For video segmentation the sure background and sure foreground regions are learned from motion statistics. SIOX features tools that allow sub-pixel accurate refinement of edges and high texture areas, the so-called "detail refinement brushes"; as with all segmentation algorithms, there are always pictures where the algorithm does not yield perfect results. The most critical drawback of SIOX is the color dependence. Although many photos are well-separable by color, the algorithm cannot deal with camouflage. If the foreground and background share many identical shades of similar colors, the algorithm might give a result with parts missing or incorrectly classified foreground.

SIOX performs about well on different benchmarks compared to graph-based segmentation methods, such as Grabcut. SIOX is, more noise robust and can therefore be used for the segmentation of videos. Graph-based segmentation methods search for a minimum cut and therefore tend to not perform optimally with complex structures; the algorithm has been developed at the department of computer science at Freie Universitaet Berlin. The main developer, Gerald Friedland, is now faculty at the EECS department of the University of California at Berkeley and a Principal Data Scientist at Lawrence Livermore National Lab, he continues to support the development through e.g. in the Google Summer of Code. G. Friedland, K. Jantz, R. Rojas: SIOX: Simple Interactive Object Extraction in Still Images, Proceedings of the IEEE International Symposium on Multimedia, pp. 253–259, December 2005. Online article G. Friedland, K. Jantz, T. Lenz, F. Wiesel, R. Rojas: Object Cut and Paste in Images and Videos, International Journal of Semantic Computing Vol 1, No 2, pp. 221–247, World Scientific, USA, June 2007.

Online article www.siox.org SIOX in GIMP Demo Video SIOX in Fiji and ImageJ www.gimp.org www.inkscape.org www.echalk.de

Hofbieber

Hofbieber is a municipality in the district of Fulda, in Hesse, Germany. Hofbieber is situated in the center of the Hessian Rhön Mountains near the Mountain Milseburg; the municipality of Hofbieber has 6,500 citizens living within eighteen urban districts and some outlying lonesome farms widespread over 90 km². The main income of the city is farming and tourism from weekend visitors from the Frankfurt Rhine-Main Region; the urban districts are: Allmus Danzwiesen Elters Hofbieber Kleinsassen Langenberg Langenbieber Mahlerts Mittelberg Niederbieber Obergruben Obernüst Rödergrund/Egelmes Schackau Schwarzbach Steens Wiesen Wittges The picturesque surroundings and a lot of well indicated hiking & biking trails are the credits of these region

2019 San Antonio FC season

The 2019 San Antonio FC season was the club's fourth season of existence. Including the San Antonio Thunder of the original NASL and the former San Antonio Scorpions of the modern NASL, this was the 10th season of professional soccer in San Antonio; the club played in the USL Championship, the second division of the United States soccer league system, participated in the U. S. Open Cup; as of 13 September 2019. The pre-season matches were announced on 14 January 2019 by SAFC. Position in the Western Conference Updated to match played 21 October 2019Source: Competitions Updated to match played 21 October 2019Source: Competitions Position in the Western Conference The first matches of 2019 were announced on 14 December 2018; the remaining schedule was released on 19 December 2018. Home team is listed first. Win Draw Loss Kickoff times are in CDT unless shown otherwise On 31 May 2019, it was announced that San Antonio would play an exhibition match against Cardiff City F. C.. Discipline includes league and Open Cup play.

The list is sorted by shirt number. The list is sorted by shirt number when total appearances are equal

Eastern river cooter

The eastern river cooter is a subspecies of turtle native to the eastern United States, with a smaller population in the midwest. It is found in freshwater habitats such as rivers and ponds; the eastern river cooter is a subspecies within the species Pseudemys concinna, known as the river cooter. The exact number of subspecies is debated, but most experts recognize two: P. c. concinna, P. c. suwanniensis. Sometimes another subspecies, P. c. floridana is recognized, but this is treated as a separate species. Eastern river cooters are capable of growing up to 16.5 inches. The carapace is dark greenish-brown with a "C" marking facing the posterior. In western populations, the "C" may be reduced and many yellow markings may be present on each scute The background color is reddish-brown, unlike the other subspecies, P. c. suwanniensis, dark. The plastron is yellow to reddish-orange with a dark pattern between scutes that follows the scute seams; this distinguishes it from P. floridana. The stripe down the hind foot is a major characteristic, P. suwanniensis can be distinguished by its lack of color on the legs.

Females tend to grow larger than males, have a smaller tail and more convex plastron. One particular distinctive feature of the Eastern River Cooter is that they have the ability to breathe underwater through a sac called the cloaca bursae, based in their tail; this allows them to stay underwater for extended periods of time, makes their behavior harder to study. Eastern river cooters prefer areas with flowing water, such as rivers, but will live in other freshwater habitats, they live in shallow areas with aquatic vegetation, when in larger numbers, they live in deeper, clear water. In the wild they feed exclusively on aquatic macrophytes and algae. Aquatic plants seem to make up 95% of their diets. Younger ones tend to seek a more protein enriched diet such as aquatic invertebrates and fish. Older turtles may seek prey as well, but partake of a herbivorous diet; these turtles can sometimes be found basking in the sun, but are wary and will retreat into the water if approached. Otherwise, they are difficult to find in the water, which may be due to their ability to breathe while submerged.

As a result, they are not seen. In warmer climates, they are active year-round, but are not active during winter in colder areas. Eastern river cooter mating habits are similar to the red-eared slider; as with the other basking turtles, the males tend to be smaller than females. The male uses his long claws to flutter at the face of the much larger female; the female ignores him. If the female is receptive, she will sink to the bottom of the river and allow the male to mount for mating. If they do mate, after several weeks the female crawls upon land to seek a nesting site, they cross highways looking for suitable nesting spots. Females will lay between 20 eggs at a time, close to water; the eggs hatch within 45 to 56 days and the hatchlings will stay with the nest through their first winter. Mating takes place in early spring. Nesting occurs from May to June; the female chooses a site with loamy soil, within 100 ft of the river's edge. She looks for a rather open area, with no major obstacles for the future hatchings to negotiate on their way to the river.

The nest is dug with the hind feet. She lays 10–25 or more eggs in one or more clutches. Eggs are ellipsoidal 1.5 inches long. Incubation time is determined by temperature, but averages 90–100 days. Hatchlings emerge in August or September. There have been reported instances of late clutches hatching in the spring. A hatchling will have a round carapace, about 1.5 inches diameter, green with bright yellow markings