Compound Poisson distribution

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https://en.wikipedia.org/wiki/Compound_Poisson_distribution

In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. The result can be either a continuous or a discrete distribution.

Definition

Suppose that

${\displaystyle N\sim \mathrm{Poisson}(\lambda ),}$

i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and that

${\displaystyle X_1,X_2,X_3,\dots }$

are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum of ${\displaystyle N}$ i.i.d. random variables

${\displaystyle Y=\sum _{n=1}^N{X}_n}$

is a compound Poisson distribution.

In the case N = 0, then this is a sum of 0 terms, so the value of Y is 0. Hence the conditional distribution of Y given that N = 0 is a degenerate distribution.

The compound Poisson distribution is obtained by marginalising the joint distribution of (Y,N) over N, and this joint distribution can be obtained by combining the conditional distribution Y | N with the marginal distribution of N.

Properties

The expected value and the variance of the compound distribution can be derived in a simple way from law of total expectation and the law of total variance. Thus

${\displaystyle \mathrm{E}(Y)=\mathrm{E}\left[\mathrm{E}(Y\mid N)\right]=\mathrm{E}\left[N\mathrm{E}(X)\right]=\mathrm{E}(N)\mathrm{E}(X),}$

${\displaystyle \begin{array}{rl}\mathrm{Var}(Y)& =\mathrm{E}\left[\mathrm{Var}(Y\mid N)\right]+\mathrm{Var}\left[\mathrm{E}(Y\mid N)\right]=\mathrm{E}\left[N\mathrm{Var}(X)\right]+\mathrm{Var}\left[N\mathrm{E}(X)\right],\\ & =\mathrm{E}(N)\mathrm{Var}(X)+{\left(\mathrm{E}(X)\right)}^2\mathrm{Var}(N).\end{array}}$

Then, since E(N) = Var(N) if N is Poisson-distributed, these formulae can be reduced to

${\displaystyle \mathrm{E}(Y)=\mathrm{E}(N)\mathrm{E}(X),}$

${\displaystyle \mathrm{Var}(Y)=\mathrm{E}(N)(\mathrm{Var}(X)+(\mathrm{E}(X){)}^2)=\mathrm{E}(N)\mathrm{E}(X^2).}$

The probability distribution of Y can be determined in terms of characteristic functions:

${\displaystyle {\phi }_Y(t)=\mathrm{E}(e^{itY})=\mathrm{E}\left({\left(\mathrm{E}(e^{itX}\mid N)\right)}^N\right)=\mathrm{E}\left(({\phi }_X(t){)}^N\right),}$

and hence, using the probability-generating function of the Poisson distribution, we have

${\displaystyle {\phi }_Y(t)={\text{e}}^{\lambda ({\phi }_X(t)-1)}.}$

An alternative approach is via cumulant generating functions:

${\displaystyle K_Y(t)=\ln \mathrm{E}[e^{tY}]=\ln \mathrm{E}[\mathrm{E}[e^{tY}\mid N]]=\ln \mathrm{E}[e^{N{K}_X(t)}]=K_N(K_X(t)).}$

Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution λ = 1, the cumulants of Y are the same as the moments of X₁.

It can be shown that every infinitely divisible probability distribution is a limit of compound Poisson distributions. And compound Poisson distributions is infinitely divisible by the definition.

Discrete compound Poisson distribution

When ${\displaystyle X_1,X_2,X_3,\dots }$ are positive integer-valued i.i.d random variables with ${\displaystyle P(X_1=k)={\alpha }_k,(k=1,2,\dots )}$, then this compound Poisson distribution is named discrete compound Poisson distribution (or stuttering-Poisson distribution) . We say that the discrete random variable ${\displaystyle Y}$ satisfying probability generating function characterization

${\displaystyle P_Y(z)=\sum _{i=0}^{\mathrm{\infty }}P(Y=i)z^i=\exp \left(\sum _{k=1}^{\mathrm{\infty }}{\alpha }_k\lambda (z^k-1)\right),(|z|\le 1)}$

has a discrete compound Poisson(DCP) distribution with parameters ${\displaystyle ({\alpha }_1\lambda ,{\alpha }_2\lambda ,\dots )\in {\mathbb{R}}^{\mathrm{\infty }}}$ (where $\sum _{i=1}^{\mathrm{\infty }}{\alpha }_i=1$, with ${\alpha }_i\ge 0,\lambda >0$), which is denoted by

${\displaystyle X\sim \text{DCP}(\lambda {\alpha }_1,\lambda {\alpha }_2,\dots )}$

Moreover, if ${\displaystyle X\sim \mathrm{DCP}(\lambda {\alpha }_1,\dots ,\lambda {\alpha }_r)}$, we say ${\displaystyle X}$ has a discrete compound Poisson distribution of order ${\displaystyle r}$ . When ${\displaystyle r=1,2}$, DCP becomes Poisson distribution and Hermite distribution, respectively. When ${\displaystyle r=3,4}$, DCP becomes triple stuttering-Poisson distribution and quadruple stuttering-Poisson distribution, respectively. Other special cases include: shift geometric distribution, negative binomial distribution, Geometric Poisson distribution, Neyman type A distribution, Luria–Delbrück distribution in Luria–Delbrück experiment. For more special case of DCP, see the reviews paper and references therein.

Feller's characterization of the compound Poisson distribution states that a non-negative integer valued r.v. ${\displaystyle X}$ is infinitely divisible if and only if its distribution is a discrete compound Poisson distribution. It can be shown that the negative binomial distribution is discrete infinitely divisible, i.e., if X has a negative binomial distribution, then for any positive integer n, there exist discrete i.i.d. random variables X₁, ..., X_n whose sum has the same distribution that X has. The shift geometric distribution is discrete compound Poisson distribution since it is a trivial case of negative binomial distribution.

This distribution can model batch arrivals (such as in a bulk queue). The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total claim amount.

When some ${\displaystyle {\alpha }_k}$ are negative, it is the discrete pseudo compound Poisson distribution. We define that any discrete random variable ${\displaystyle Y}$ satisfying probability generating function characterization

${\displaystyle G_Y(z)=\sum _{i=0}^{\mathrm{\infty }}P(Y=i)z^i=\exp \left(\sum _{k=1}^{\mathrm{\infty }}{\alpha }_k\lambda (z^k-1)\right),(|z|\le 1)}$

has a discrete pseudo compound Poisson distribution with parameters ${\displaystyle ({\lambda }_1,{\lambda }_2,\dots )=:({\alpha }_1\lambda ,{\alpha }_2\lambda ,\dots )\in {\mathbb{R}}^{\mathrm{\infty }}}$ where $\sum _{i=1}^{\mathrm{\infty }}{\alpha }_i=1$ and $\sum _{i=1}^{\mathrm{\infty }}\left|{\alpha }_i\right|<\mathrm{\infty }$, with ${\displaystyle {\alpha }_i\in \mathbb{R},\lambda >0}$.

Compound Poisson Gamma distribution

If X has a gamma distribution, of which the exponential distribution is a special case, then the conditional distribution of Y | N is again a gamma distribution. The marginal distribution of Y can be shown to be a Tweedie distribution with variance power 1 < p < 2 (proof via comparison of characteristic function (probability theory)). To be more explicit, if

${\displaystyle N\sim \mathrm{Poisson}(\lambda ),}$

and

${\displaystyle X_i\sim \mathrm{\Gamma }(\alpha ,\beta )}$

i.i.d., then the distribution of

${\displaystyle Y=\sum _{i=1}^N{X}_i}$

is a reproductive exponential dispersion model ${\displaystyle ED(\mu ,{\sigma }^2)}$ with

${\displaystyle \begin{array}{rl}\mathrm{E}[Y]& =\lambda \frac{\alpha }{\beta }=:\mu ,\\ \mathrm{Var}[Y]& =\lambda \frac{\alpha (1+\alpha )}{{\beta }^2}=:{\sigma }^2{\mu }^p.\end{array}}$

The mapping of parameters Tweedie parameter ${\displaystyle \mu ,{\sigma }^2,p}$ to the Poisson and Gamma parameters ${\displaystyle \lambda ,\alpha ,\beta }$ is the following:

${\displaystyle \begin{array}{rl}\lambda & =\frac{{\mu }^{2-p}}{(2-p){\sigma }^2},\\ \alpha & =\frac{2-p}{p-1},\\ \beta & =\frac{{\mu }^{1-p}}{(p-1){\sigma }^2}.\end{array}}$

Compound Poisson processes

Main article: Compound Poisson process

A compound Poisson process with rate ${\displaystyle \lambda >0}$ and jump size distribution G is a continuous-time stochastic process ${\displaystyle \{Y(t):t\ge 0\}}$ given by

${\displaystyle Y(t)=\sum _{i=1}^{N(t)}{D}_i,}$

where the sum is by convention equal to zero as long as N(t) = 0. Here, ${\displaystyle \{N(t):t\ge 0\}}$ is a Poisson process with rate ${\displaystyle \lambda }$, and ${\displaystyle \{D_i:i\ge 1\}}$ are independent and identically distributed random variables, with distribution function G, which are also independent of ${\displaystyle \{N(t):t\ge 0\}.}$

For the discrete version of compound Poisson process, it can be used in survival analysis for the frailty models.

Applications

A compound Poisson distribution, in which the summands have an exponential distribution, was used by Revfeim to model the distribution of the total rainfall in a day, where each day contains a Poisson-distributed number of events each of which provides an amount of rainfall which has an exponential distribution. Thompson applied the same model to monthly total rainfalls.

There have been applications to insurance claims and x-ray computed tomography.

Pharynx

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Pharynx 

 

Pharynx
Head and inner neck
Pharynx
Details
Part of	Throat
System	Respiratory system, digestive system
Artery	pharyngeal branches of ascending pharyngeal artery, ascending palatine, descending palatine, pharyngeal branches of inferior thyroid
Vein	pharyngeal plexus
Nerve	pharyngeal plexus of vagus nerve, recurrent laryngeal nerve, maxillary nerve, mandibular nerve

The pharynx (PL: pharynges) is the part of the throat behind the mouth and nasal cavity, and above the esophagus and trachea (the tubes going down to the stomach and the lungs respectively). It is found in vertebrates and invertebrates, though its structure varies across species. The pharynx carries food to the esophagus and air to the larynx. The flap of cartilage called the epiglottis stops food from entering the larynx.

In humans, the pharynx is part of the digestive system and the conducting zone of the respiratory system. (The conducting zone—which also includes the nostrils of the nose, the larynx, trachea, bronchi, and bronchioles—filters, warms and moistens air and conducts it into the lungs). The human pharynx is conventionally divided into three sections: the nasopharynx, oropharynx, and laryngopharynx. It is also important in

In humans, two sets of pharyngeal muscles form the pharynx and determine the shape of its lumen. They are arranged as an inner layer of longitudinal muscles and an outer circular layer.

Structure

Nasopharynx

Upper respiratory system, with the nasopharynx, oropharynx and laryngopharynx labeled at left

The upper portion of the pharynx, the nasopharynx, extends from the base of the skull to the upper surface of the soft palate. It includes the space between the internal nares and the soft palate and lies above the oral cavity. The adenoids, also known as the pharyngeal tonsils, are lymphoid tissue structures located in the posterior wall of the nasopharynx. Waldeyer's tonsillar ring is an annular arrangement of lymphoid tissue in both the nasopharynx and oropharynx. The nasopharynx is lined by respiratory epithelium that is pseudostratified, columnar, and ciliated.

Polyps or mucus can obstruct the nasopharynx, as can congestion due to an upper respiratory infection. The auditory tube, which connects the middle ear to the pharynx, opens into the nasopharynx at the pharyngeal opening of the auditory tube. The opening and closing of the auditory tubes serves to equalize the barometric pressure in the middle ear with that of the ambient atmosphere.

Details of torus tubarius

The anterior aspect of the nasopharynx communicates through the choanae with the nasal cavities. On its lateral wall is the pharyngeal opening of the auditory tube, somewhat triangular in shape and bounded behind by a firm prominence, the torus tubarius or cushion, caused by the medial end of the cartilage of the tube that elevates the mucous membrane. Two folds arise from the cartilaginous opening:

• the salpingopharyngeal fold, a vertical fold of mucous membrane extending from the inferior part of the torus and containing the salpingopharyngeus muscle
• the salpingopalatine fold, a smaller fold, in front of the salpingopharyngeal fold, extending from the superior part of the torus to the palate and containing the levator veli palatini muscle. It also contains some muscle fibres called salpingopalatine muscle The tensor veli palatini is lateral to the levator and does not contribute to the fold, since the origin is deep to the cartilaginous opening.

Oropharynx

The oropharynx lies behind the oral cavity, extending from the uvula to the level of the hyoid bone. It opens anteriorly, through the isthmus faucium, into the mouth, while in its lateral wall, between the palatoglossal arch and the palatopharyngeal arch, is the palatine tonsil. The anterior wall consists of the base of the tongue and the epiglottic vallecula; the lateral wall is made up of the tonsil, tonsillar fossa, and tonsillar (faucial) pillars; the superior wall consists of the inferior surface of the soft palate and the uvula. Because both food and air pass through the pharynx, a flap of connective tissue called the epiglottis closes over the glottis when food is swallowed to prevent aspiration. The oropharynx is lined by non-keratinized squamous stratified epithelium.

The HACEK organisms (Haemophilus, Actinobacillus actinomycetemcomitans, Cardiobacterium hominis, Eikenella corrodens, Kingella) are part of the normal oropharyngeal flora, which grow slowly, prefer a carbon dioxide-enriched atmosphere, and share an enhanced capacity to produce endocardial infections, especially in young children. Fusobacterium is a pathogen.

Laryngopharynx

The laryngopharynx, (Latin: pars laryngea pharyngis), also known as hypopharynx, is the caudal part of the pharynx; it is the part of the throat that connects to the esophagus. It lies inferior to the epiglottis and extends to the location where this common pathway diverges into the respiratory (laryngeal) and digestive (esophageal) pathways. At that point, the laryngopharynx is continuous with the esophagus posteriorly. The esophagus conducts food and fluids to the stomach; air enters the larynx anteriorly. During swallowing, food has the "right of way", and air passage temporarily stops. Corresponding roughly to the area located between the 4th and 6th cervical vertebrae, the superior boundary of the laryngopharynx is at the level of the hyoid bone. The laryngopharynx includes three major sites: the pyriform sinus, postcricoid area, and the posterior pharyngeal wall. Like the oropharynx above it, the laryngopharynx serves as a passageway for food and air and is lined with a stratified squamous epithelium. It is innervated by the pharyngeal plexus and by the recurrent laryngeal nerve.

The vascular supply to the laryngopharynx includes the superior thyroid artery, the lingual artery and the ascending pharyngeal artery. The primary neural supply is from both the vagus and glossopharyngeal nerves. The vagus nerve provides an auricular branch also termed "Arnold's nerve" which also supplies the external auditory canal, thus laryngopharyngeal cancer can result in referred ear pain. This nerve is also responsible for the ear-cough reflex in which stimulation of the ear canal results in a person coughing.

Function

The pharynx moves food from the mouth to the esophagus. It also moves air from the nasal and oral cavities to the larynx. It is also used in human speech, as pharyngeal consonants are articulated here, and it acts as a resonating chamber during phonation.

Clinical significance

Pharyngitis is the painful swelling of the throat. The oropharynx shown here is very inflamed and red.

Inflammation

Main article: Pharyngitis

Inflammation of the pharynx, or pharyngitis, is the painful inflammation of the throat.

Pharyngeal cancer

Main article: Pharyngeal cancer

Pharyngeal cancer is a cancer that originates in the neck and/or throat.

Waldeyer's tonsillar ring

Main article: Waldeyer's tonsillar ring

Waldeyer's tonsillar ring is an anatomical term collectively describing the annular arrangement of lymphoid tissue in the pharynx. Waldeyer's ring circumscribes the naso- and oropharynx, with some of its tonsillar tissue located above and some below the soft palate (and to the back of the oral cavity). It is believed that Waldeyer's ring prevents the invasion of microorganisms from going into the air and food passages and this helps in the defense mechanism of the respiratory and alimentary systems.

Etymology

The word pharynx (/ˈfærɪŋks/) is derived from the Greek φάρυγξ phárynx, meaning "throat". Its plural form is pharynges /fəˈrɪndʒiːz/ or pharynxes /ˈfærɪŋksəz/, and its adjective form is pharyngeal (/ˌfærɪnˈdʒiːəl/ or /fəˈrɪndʒiəl/).

Other vertebrates

All vertebrates have a pharynx, used in both feeding and respiration. The pharynx arises during development in all vertebrates through a series of six or more outpocketings on the lateral sides of the head. These outpocketings are pharyngeal arches, and they give rise to a number of different structures in the skeletal, muscular, and circulatory systems. The structure of the pharynx varies across the vertebrates. It differs in dogs, horses, and ruminants. In dogs, a single duct connects the nasopharynx to the nasal cavity. The tonsils are a compact mass that points away from the lumen of the pharynx. In the horse, the auditory tube opens into the guttural pouch and the tonsils are diffuse and raised slightly. Horses are unable to breathe through the mouth as the free apex of the rostral epiglottis lies dorsal to the soft palate in a normal horse. In ruminants the tonsils are a compact mass that points towards the lumen of the pharynx.

Pharyngeal arches

Pharyngeal arches are characteristic features of vertebrates whose origin can be traced back through chordates to basal deuterostomes who also share endodermal outpocketings of the pharyngeal apparatus. Similar patterns of gene expression can be detected in the developing pharynx of amphioxi and hemichordates. However, the vertebrate pharynx is unique in that it gives rise to endoskeletal support through the contribution of neural crest cells.

Pharyngeal jaws

An illustration of the pharyngeal jaws of a moray eel

Pharyngeal jaws are a "second set" of jaws contained within the pharynx of many species of fish, distinct from the primary (oral) jaws. Pharyngeal jaws have been studied in moray eels where their specific action is noted. When the moray bites prey, it first bites normally with its oral jaws, capturing the prey. Immediately thereafter, the pharyngeal jaws are brought forward and bite down on the prey to grip it; they then retract, pulling the prey down the eel's esophagus, allowing it to be swallowed.

Invertebrates

Invertebrates also have a pharynx. Invertebrates with a pharynx include the tardigrades, annelids and arthropods, and the priapulids (which have an eversible pharynx).

The "pharynx" of the nematode worm is a muscular food pump in the head, triangular in cross-section, that grinds food and transports it directly to the intestines. A one-way valve connects the pharynx to the excretory canal.

Noble gas compound

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Noble_gas_compound

In chemistry, noble gas compounds are chemical compounds that include an element from the noble gases, group 18 of the periodic table. Although the noble gases are generally unreactive elements, many such compounds have been observed, particularly involving the element xenon.

From the standpoint of chemistry, the noble gases may be divided into two groups: the relatively reactive krypton (ionisation energy 14.0 eV), xenon (12.1 eV), and radon (10.7 eV) on one side, and the very unreactive argon (15.8 eV), neon (21.6 eV), and helium (24.6 eV) on the other. Consistent with this classification, Kr, Xe, and Rn form compounds that can be isolated in bulk at or near standard temperature and pressure, whereas He, Ne, Ar have been observed to form true chemical bonds using spectroscopic techniques, but only when frozen into a noble gas matrix at temperatures of 40 K or lower, in supersonic jets of noble gas, or under extremely high pressures with metals.

The heavier noble gases have more electron shells than the lighter ones. Hence, the outermost electrons are subject to a shielding effect from the inner electrons that makes them more easily ionized, since they are less strongly attracted to the positively-charged nucleus. This results in an ionization energy low enough to form stable compounds with the most electronegative elements, fluorine and oxygen, and even with less electronegative elements such as nitrogen and carbon under certain circumstances.

History and background

When the family of noble gases was first identified at the end of the nineteenth century, none of them were observed to form any compounds and so it was initially believed that they were all inert gases (as they were then known) which could not form compounds. With the development of atomic theory in the early twentieth century, their inertness was ascribed to a full valence shell of electrons which render them very chemically stable and nonreactive. All noble gases have full s and p outer electron shells (except helium, which has no p sublevel), and so do not form chemical compounds easily. Their high ionization energy and almost zero electron affinity explain their non-reactivity.

In 1933, Linus Pauling predicted that the heavier noble gases would be able to form compounds with fluorine and oxygen. Specifically, he predicted the existence of krypton hexafluoride (KrF₆) and xenon hexafluoride (XeF₆), speculated that XeF₈ might exist as an unstable compound, and suggested that xenic acid would form perxenate salts. These predictions proved quite accurate, although subsequent predictions for XeF₈ indicated that it would be not only thermodynamically unstable, but kinetically unstable. As of 2022, XeF₈ has not been made, although the octafluoroxenate(VI) anion ([XeF₈]²⁻) has been observed.

By 1960, no compound with a covalently bound noble gas atom had yet been synthesized. The first published report, in June 1962, of a noble gas compound was by Neil Bartlett, who noticed that the highly oxidising compound platinum hexafluoride ionised O₂ to O+2. As the ionisation energy of O₂ to O+2 (1165 kJ mol⁻¹) is nearly equal to the ionisation energy of Xe to Xe⁺ (1170 kJ mol⁻¹), he tried the reaction of Xe with PtF₆. This yielded a crystalline product, xenon hexafluoroplatinate, whose formula was proposed to be Xe⁺[PtF₆]⁻. It was later shown that the compound is actually more complex, containing both [XeF]⁺[PtF₅]⁻ and [XeF]⁺[Pt₂F₁₁]⁻. Nonetheless, this was the first real compound of any noble gas.

The first binary noble gas compounds were reported later in 1962. Bartlett synthesized xenon tetrafluoride (XeF₄) by subjecting a mixture of xenon and fluorine to high temperature. Rudolf Hoppe, among other groups, synthesized xenon difluoride (XeF₂) by the reaction of the elements.

Following the first successful synthesis of xenon compounds, synthesis of krypton difluoride (KrF₂) was reported in 1963.

True noble gas compounds

In this section, the non-radioactive noble gases are considered in decreasing order of atomic weight, which generally reflects the priority of their discovery, and the breadth of available information for these compounds. The radioactive elements radon and oganesson are harder to study and are considered at the end of the section.

Xenon compounds

Main article: Xenon compounds

After the initial 1962 studies on XeF₄ and XeF₂, xenon compounds that have been synthesized include other fluorides (XeF₆), oxyfluorides (XeOF₂, XeOF₄, XeO₂F₂, XeO₃F₂, XeO₂F₄) and oxides (XeO₂, XeO₃ and XeO₄). Xenon fluorides react with several other fluorides to form fluoroxenates, such as sodium octafluoroxenate(VI) ((Na⁺)₂[XeF₈]²⁻), and fluoroxenonium salts, such as trifluoroxenonium hexafluoroantimonate ([XeF₃]⁺[SbF₆]⁻).

In terms of other halide reactivity, short-lived excimers of noble gas halides such as XeCl₂ or XeCl are prepared in situ, and are used in the function of excimer lasers.

Recently, xenon has been shown to produce a wide variety of compounds of the type XeO_nX₂ where n is 1, 2 or 3 and X is any electronegative group, such as CF₃, C(SO₂CF₃)₃, N(SO₂F)₂, N(SO₂CF₃)₂, OTeF₅, O(IO₂F₂), etc.; the range of compounds is impressive, similar to that seen with the neighbouring element iodine, running into the thousands and involving bonds between xenon and oxygen, nitrogen, carbon, boron and even gold, as well as perxenic acid, several halides, and complex ions.

The compound [Xe₂]⁺[Sb₄F₂₁]⁻ contains a Xe–Xe bond, which is the longest element-element bond known (308.71 pm = 3.0871 Å). Short-lived excimers of Xe₂ are reported to exist as a part of the function of excimer lasers.

Krypton compounds

Main article: Krypton § Chemistry

Krypton gas reacts with fluorine gas under extreme forcing conditions, forming KrF₂ according to the following equation:

Kr + F₂ → KrF₂

KrF₂ reacts with strong Lewis acids to form salts of the [KrF]⁺ and [Kr₂F₃]⁺ cations. The preparation of KrF₄ reported by Grosse in 1963, using the Claasen method, was subsequently shown to be a mistaken identification.

Krypton compounds with other than Kr–F bonds (compounds with atoms other than fluorine) have also been described. KrF₂ reacts with B(OTeF₅)₃ to produce the unstable compound, Kr(OTeF₅)₂, with a krypton-oxygen bond. A krypton-nitrogen bond is found in the cation [H−C≡N−Kr−F]⁺, produced by the reaction of KrF₂ with [H−C≡N−H]⁺[AsF₆]⁻ below −50 °C.

Argon compounds

Main article: Argon compounds

The discovery of HArF was announced in 2000. The compound can exist in low temperature argon matrices for experimental studies, and it has also been studied computationally. Argon hydride ion [ArH]⁺ was obtained in the 1970s. This molecular ion has also been identified in the Crab nebula, based on the frequency of its light emissions.

There is a possibility that a solid salt of [ArF]⁺ could be prepared with [SbF₆]⁻ or [AuF₆]⁻ anions.

Neon and helium compounds

Main articles: Helium compounds and Neon compounds

The ions, Ne⁺, [NeAr]⁺, [NeH]⁺, and [HeNe]⁺ are known from optical and mass spectrometric studies. Neon also forms an unstable hydrate. There is some empirical and theoretical evidence for a few metastable helium compounds which may exist at very low temperatures or extreme pressures. The stable cation [HeH]⁺ was reported in 1925, but was not considered a true compound since it is not neutral and cannot be isolated. In 2016 scientists created the helium compound disodium helide (Na₂He) which was the first helium compound discovered.

Radon and oganesson compounds

Main articles: Radon § Chemical properties, and Oganesson § Predicted compounds

Radon is not chemically inert, but its short half-life (3.8 days for ²²²Rn) and the high energy of its radioactivity make it difficult to investigate its only fluoride (RnF₂), its reported oxide (RnO₃), and their reaction products.

All known oganesson isotopes have even shorter half-lives in the millisecond range and no compounds are known yet, although some have been predicted theoretically. It is expected to be even more reactive than radon, more like a normal element than a noble gas in its chemistry.

Reports prior to xenon hexafluoroplatinate and xenon tetrafluoride

Clathrates

Kr(H₂)₄ and H₂ solids formed in a diamond anvil cell. Ruby was added for pressure measurement.

Structure of Kr(H₂)₄. Krypton octahedra (green) are surrounded by randomly oriented hydrogen molecules.

Prior to 1962, the only isolated compounds of noble gases were clathrates (including clathrate hydrates); other compounds such as coordination compounds were observed only by spectroscopic means. Clathrates (also known as cage compounds) are compounds of noble gases in which they are trapped within cavities of crystal lattices of certain organic and inorganic substances. The essential condition for their formation is that the guest (noble gas) atoms should be of appropriate size to fit in the cavities of the host crystal lattice; for instance, Ar, Kr, and Xe can form clathrates with crystalline β-quinol, but He and Ne cannot fit because they are too small. As well, Kr and Xe can appear as guests in crystals of melanophlogite.

Helium-nitrogen (He(N₂)₁₁) crystals have been grown at room temperature at pressures ca. 10 GPa in a diamond anvil cell. Solid argon-hydrogen clathrate (Ar(H₂)₂) has the same crystal structure as the MgZn₂ Laves phase. It forms at pressures between 4.3 and 220 GPa, though Raman measurements suggest that the H₂ molecules in Ar(H₂)₂ dissociate above 175 GPa. A similar Kr(H₂)₄ solid forms at pressures above 5 GPa. It has a face-centered cubic structure where krypton octahedra are surrounded by randomly oriented hydrogen molecules. Meanwhile, in solid Xe(H₂)₈ xenon atoms form dimers inside solid hydrogen.

Coordination compounds

Coordination compounds such as Ar·BF₃ have been postulated to exist at low temperatures, but have never been confirmed. Also, compounds such as WHe₂ and HgHe₂ were reported to have been formed by electron bombardment, but recent research has shown that these are probably the result of He being adsorbed on the surface of the metal; therefore, these compounds cannot truly be considered chemical compounds.

Hydrates

Hydrates are formed by compressing noble gases in water, where it is believed that the water molecule, a strong dipole, induces a weak dipole in the noble gas atoms, resulting in dipole-dipole interaction. Heavier atoms are more influenced than smaller ones, hence Xe·5.75H₂O was reported to have been the most stable hydrate; it has a melting point of 24 °C. The deuterated version of this hydrate has also been produced.

Fullerene adducts

Main article: Non-metal doped fullerenes

Structure of a noble-gas atom caged within a buckminsterfullerene (C₆₀) molecule.

Noble gases can also form endohedral fullerene compounds where the noble gas atom is trapped inside a fullerene molecule. In 1993, it was discovered that when C₆₀ is exposed to a pressure of around 3 bar of He or Ne, the complexes He@C₆₀ and Ne@C₆₀ are formed. Under these conditions, only about one out of every 650,000 C₆₀ cages was doped with a helium atom; with higher pressures (3000 bar), it is possible to achieve a yield of up to 0.1%. Endohedral complexes with argon, krypton and xenon have also been obtained, as well as numerous adducts of He@C₆₀.

Applications

Most applications of noble gas compounds are either as oxidising agents or as a means to store noble gases in a dense form. Xenic acid is a valuable oxidising agent because it has no potential for introducing impurities—xenon is simply liberated as a gas—and so is rivalled only by ozone in this regard. The perxenates are even more powerful oxidizing agents. Xenon-based oxidants have also been used for synthesizing carbocations stable at room temperature, in SO₂ClF solution.

Stable salts of xenon containing very high proportions of fluorine by weight (such as tetrafluoroammonium heptafluoroxenate(VI), [NF₄][XeF₇], and the related tetrafluoroammonium octafluoroxenate(VI) [NF₄]₂[XeF₈]), have been developed as highly energetic oxidisers for use as propellants in rocketry.

Xenon fluorides are good fluorinating agents.

Clathrates have been used for separation of He and Ne from Ar, Kr, and Xe, and also for the transportation of Ar, Kr, and Xe. (For instance, radioactive isotopes of krypton and xenon are difficult to store and dispose, and compounds of these elements may be more easily handled than the gaseous forms.) In addition, clathrates of radioisotopes may provide suitable formulations for experiments requiring sources of particular types of radiation; hence. ⁸⁵Kr clathrate provides a safe source of beta particles, while ¹³³Xe clathrate provides a useful source of gamma rays.