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Probability density function

In probability theory, a probability density function, or density of a continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more it is that the random variable would equal one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value; this probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range.

The probability density function is nonnegative everywhere, its integral over the entire space is equal to 1. The terms "probability distribution function" and "probability function" have sometimes been used to denote the probability density function. However, this use is not standard among statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function, or it may be a probability mass function rather than the density. "Density function" itself is used for the probability mass function, leading to further confusion. In general though, the PMF is used in the context of discrete random variables, while the PDF is used in the context of continuous random variables. Suppose bacteria of a certain species live 4 to 6 hours; the probability that a bacterium lives 5 hours is equal to zero. A lot of bacteria live for 5 hours, but there is no chance that any given bacterium dies at 5.0000000000... hours.

However, the probability that the bacterium dies between 5 hours and 5.01 hours is quantifiable. Suppose the answer is 0.02. The probability that the bacterium dies between 5 hours and 5.001 hours should be about 0.002, since this time interval is one-tenth as long as the previous. The probability that the bacterium dies between 5 hours and 5.0001 hours should be about 0.0002, so on. In these three examples, the ratio / is constant, equal to 2 per hour. For example, there is 0.02 probability of dying in the 0.01-hour interval between 5 and 5.01 hours, = 2 hour−1. This quantity 2 hour−1 is called the probability density for dying at around 5 hours. Therefore, the probability that the bacterium dies at 5 hours can be written as dt; this is the probability that the bacterium dies within an infinitesimal window of time around 5 hours, where dt is the duration of this window. For example, the probability that it lives longer than 5 hours, but shorter than, is × ≈ 6×10−13. There is a probability density function f with f = 2 hour−1.

The integral of f over any window of time is the probability. A probability density function is most associated with continuous univariate distributions. A random variable X has density f X, where f X is a non-negative Lebesgue-integrable function, if: Pr = ∫ a b f X d x. Hence, if F X is the cumulative distribution function of X, then: F X = ∫ − ∞ x f X d u, f X = d d x F X. Intuitively, one can think of f X d x as being the probability of X falling within the infinitesimal interval. A random variable X with values in a measurable space (usually R n {\displaystyle

Jabez Gough Enclosure

The Jabez Gough Enclosure was invented in 1960 by Jabez Gough, a radio engineer, living in Cardiff, South Wales. Gough devised and constructed a wooden housing for a loudspeaker unit which also acted as an acoustic chamber, rather in the way that the body of a violin serves as a sound box for the strings. Challenging the dominant acoustic theories of his day and opposed by some major loudspeaker manufacturers, Gough decided to publish and sell his own plans for the construction of the enclosure, which proved popular in this do-it-yourself era and many thousands of copies were purchased by hi-fi enthusiasts worldwide; the Jabez Gough Enclosure was first demonstrated publicly on 19 October 1960, in Tongwynlais, South Wales. The event attracted national publicity. In a front-page article for The Observer on 27 November, Peter Schirmer declared that a Gough enclosure, fitted with a single 12-inch loudspeaker unit, "could fill St Pauls Cathedral with perfect high fidelity sound." Gough subsequently took a pair of his enclosures to demonstrate in London's famous Covent Garden Opera House.

Powered by a single 15 watt valve amplifier, fitted with two eight-inch loudspeaker units, the Jabez Gough Enclosures managed to fill the opera house with sound. Jabez Gough patented his invention in August 1961, but no large-scale manufacture of the units took place because Gough himself was not interested in commercial development of the idea. Word of the invention spread far and wide around the world. Meanwhile, in the UK, Gough's invention was the focus for a popular BBC television programme in the'How to Make it' series, in July 1961. By 1973 some 35,000 copies of the construction plans for the enclosure had been sold around the world. One specialist commentator wrote at this time that the Gough Enclosure, its revolutionary design, was "one of the biggest controversies the hi-fi world has known."Measuring six feet square, the Jabez Gough Enclosure lost popularity as portability and miniaturisation of audio systems took hold in the eighties and nineties. Now in the modern digital era, the Gough loudspeaker is all but extinct.

However, Gough's legacy is still not quite forgotten as fifty years his work is still the subject of public controversy, as is witnessed in a recent feature in Hi-Fi News. tribute website

Red Wing Medical Center

Mayo Clinic Health System in Red Wing is a 50-bed hospital located in Red Wing, United States. Opened in 2001, it combines a hospital in one facility; the first hospital in Red Wing, Minnesota was established in 1884. Located at southwest Dakota and Levee Streets; the Andrew Koch building served as the first hospital in Red Wing. The house was built by A. Koch in the 1850s; the current hospital was formed from the merge of River Region Health Services and Interstate Medical Center in 1997. Red Wing was served by St. John's Regional Health Center and was organized into River Region Health Services in 1986, it was determined in a community-wide planning process, initiated in 1985 and was completed in mid-1986, that the city of Red Wing wanted to create a regional health care system. The board leadership of St. John's Regional Health Center, Seminary Memorial Home and Interstate Medical Center agreed with the goals of "Red Wing 2000." An organizational format was created for. Recommendations to create a new system and to build stronger working relationships with physicians were presented to the boards of St. John's Regional Health Center, Seminary Memorial Home and Interstate Medical Center in September 1986.

There was a clear vision that the future trend in health care was toward de-institutionalization––outpatient and home care, alternative living arrangements. To meet the changing needs, RRHS created two additional subsidiaries in River Region Community Services and River Region Housing Corporation. Interstate Medical Center was a professional corporation and its predecessor was the Interstate Clinic, it was founded in a partnership between Edward H. Juers, M. D. and Raymond F. Hedin, M. D. in 1932. In 1940, having additional physicians join the medical staff, their group, now the Interstate Clinic, moved to a new building at Third and Dakota streets; the group incorporated as the Interstate Medical Center, Professional Associates in 1969, moved to US Highway 61 in 1970. A major expansion of the building occurred in 1980. In 1986, a second building was added at this site. In 1981, George M. B. Hawley, M. D. became associated with the Interstate Medical Center and the first branch office began operation at 303 West Fifth Street.

In 1987, that office moved to Fourth Street in the Seminary Plaza. The branch office in Ellsworth, Wisconsin was constructed in 1984. In 1986, Robert Thompson, M. D. in Zumbrota, joined Interstate Medical Center. That office moved to 525 Mill Street. In 1997, Fairview Red Wing Health Services brought together River Region Health Services and Interstate Medical Center as one coordinated system of care; the new Fairview Red Wing Medical Center served the clinic and hospital needs of Red Wing and the surrounding area. Red Wing Medical Center opened in 2001 on the West side of Red Wing, Minnesota located on Fairview Boulevard, just off Red Wing Avenue S near US Highway 61. Red Wing Medical Center was purchased by Mayo Clinic in 2012 and was renamed Mayo Clinic Health System in Red Wing. Mayo Clinic Health System in Red Wing is part of Mayo Clinic Health System, their merger and partnership with Mayo Clinic Health System and Mayo Clinic puts the former Red Wing Medical Center at the forefront of world-class research and innovative patient treatments.

The University of Minnesota is world-renowned for leading in transplants, heart surgery and treatments for other diseases. The partnership we entered in 1997 stands as a national model for academic- and community-based health system collaboration. Mayo Clinic Health System - Red Wing

USS Sailfish (SS-192)

USS Sailfish, was a US Sargo-class submarine named Squalus. As the Squalus, the submarine sank off the coast of New Hampshire during test dives on 23 May 1939; the sinking drowned 26 crew members, but an ensuing rescue operation using the McCann Rescue Chamber saved the lives of the remaining 33 aboard. The submarine was decommissioned; the submarine was recommissioned as the Sailfish in May 1940, conducted numerous patrols in the Pacific War during World War II, earning nine battle stars. She was decommissioned in October 1945 and scrapped, her keel was laid on 18 October 1937 by the Portsmouth Naval Shipyard in Kittery, Maine, as Squalus, the only ship of the United States Navy named for the squalus, a type of shark. She was launched on 14 September 1938 sponsored by Mrs. Thomas C. Hart, commissioned on 1 March 1939, with Lieutenant Oliver F. Naquin in command. Due to mechanical failure, Squalus sank during a test dive on 23 May 1939, she was raised and recommissioned on 15 May 1940 as Sailfish.

On 12 May 1939, following a yard overhaul, Squalus began a series of test dives off Portsmouth, New Hampshire. After completing 18 dives, she went down again off the Isles of Shoals on the morning of 23 May at 42°53′N 70°37′W. Failure of the main induction valve caused the flooding of the aft torpedo room, both engine rooms, the crew's quarters, drowning 26 men immediately. Quick action by the crew prevented the other compartments from flooding. Squalus bottomed in 243 ft of water. Squalus was located by her sister ship, Sculpin; the two submarines were able to communicate using a telephone marker buoy. Divers from the submarine rescue ship Falcon began rescue operations under the direction of the salvage and rescue expert Lieutenant Commander Charles B. "Swede" Momsen, using the new McCann Rescue Chamber. The Senior Medical Officer for the operations was Dr. Charles Wesley Shilling. Overseen by researcher Albert R. Behnke, the divers used developed heliox diving schedules and avoided the cognitive impairment symptoms associated with such deep dives, thereby confirming Behnke's theory of nitrogen narcosis.

The divers were able to rescue all 33 survivors on board the sunken submarine. Four enlisted divers, Chief Machinist's Mate William Badders, Chief Boatswain's Mate Orson L. Crandall, Chief Metalsmith James H. McDonald and Chief Torpedoman John Mihalowski, were awarded the Medal of Honor for their work during the rescue and subsequent salvage; the successful rescue of the Squalus survivors is in marked contrast to the loss of Thetis in Liverpool Bay just a week later.) The navy authorities felt it important to raise her as she incorporated a succession of new design features. With a thorough investigation of why she sank, more confidence could be placed in the new construction, or alteration of existing designs could be undertaken when cheapest and most efficient to do so. Furthermore, given similar previous accidents in Sturgeon and Snapper, it was necessary to determine a cause; the salvage of Squalus was commanded by Rear Admiral Cyrus W. Cole, Commander of the Portsmouth Naval Shipyard, who supervised salvage officer Lieutenant Floyd A. Tusler from the Construction Corps.

Tusler's plan was to lift the submarine in three stages to prevent it from rising too out of control, with one end up, in which case there would be a high likelihood of it sinking again. For 50 days, divers worked to attach pontoons for buoyancy. On 13 July 1939, the stern was raised but when the men attempted to free the bow from the hard blue clay, the vessel began to rise far too slipping its cables. Ascending vertically, the submarine broke the surface, 30 feet of the bow reached into the air for not more than ten seconds before she sank once again all the way to the bottom. Momsen said of the mishap, "pontoons were smashed, hoses cut and I might add, hearts were broken." After 20 more days of preparation, with a radically redesigned pontoon and cable arrangement, the next lift was successful, as were two further operations. Squalus was towed into Portsmouth on 13 September, decommissioned on 15 November. A total of 628 dives had been made in salvage operations. Renamed Sailfish on 9 February 1940, she became the first ship of the U.

S. Navy named for the sailfish. After reconditioning and overhaul, she was recommissioned on 15 May 1940 with Lieutenant Commander Morton C. Mumma, Jr. in command. With refit completed in mid-September, Sailfish departed Portsmouth on 16 January 1941 and headed for the Pacific. Transiting the Panama Canal, she arrived at Pearl Harbor in early March, after refueling at San Diego; the submarine sailed west to Manila where she joined the Asiatic Fleet until the attack on Pearl Harbor. During the Pacific War, the captain of the renamed ship issued standing orders if any man on the boat said the word "Squalus", he was to be marooned at the next port of call; this led to crew members referring to their ship as "Squailfish". That went over as well. Following the attack on Pearl Harbor, Sailfish departed Manila on her first war patrol, destined for the west coast of Luzon. Early on 10 December, she sighted a landing force, supported by cruisers and destroyers, but could not gain firing position. On the night of 13 December, she made contact with two Japanese destroyers and began a submerged attack.

Tolman–Oppenheimer–Volkoff equation

In astrophysics, the Tolman–Oppenheimer–Volkoff equation constrains the structure of a spherically symmetric body of isotropic material, in static gravitational equilibrium, as modelled by general relativity. The equation is d P d r = − G m r 2 ρ − 1 Here, r is a radial coordinate, ρ and P are the density and pressure of the material at r = r0; the quantity m, the total mass within r0, is discussed below. The equation is derived by solving the Einstein equations for a general time-invariant, spherically symmetric metric. For a solution to the Tolman–Oppenheimer–Volkoff equation, this metric will take the form d s 2 = e ν c 2 d t 2 − − 1 d r 2 − r 2 where ν is determined by the constraint d ν d r = − d P d r When supplemented with an equation of state, F = 0, which relates density to pressure, the Tolman–Oppenheimer–Volkoff equation determines the structure of a spherically symmetric body of isotropic material in equilibrium. If terms of order 1/c2 are neglected, the Tolman–Oppenheimer–Volkoff equation becomes the Newtonian hydrostatic equation, used to find the equilibrium structure of a spherically symmetric body of isotropic material when general-relativistic corrections are not important.

If the equation is used to model a bounded sphere of material in a vacuum, the zero-pressure condition P = 0 and the condition eν = 1 − 2Gm/rc2 should be imposed at the boundary. The second boundary condition is imposed so that the metric at the boundary is continuous with the unique static spherically symmetric solution to the vacuum field equations, the Schwarzschild metric: d s 2 = c 2 d t 2 − − 1 d r 2 − r 2 m is the total mass inside radius r = r0, as measured by the gravitational field felt by a distant observer, it satisfies m = 0. D m d r = 4 π r 2 ρ Here, M is the total mass of the object, again, as measured by the gravitational field felt by a distant observer. If the boundary is at r = R, continuity of the metric and the definition of m require that M = m = ∫ 0 R 4 π r 2 ρ d r Computing the mass by integrating the density of the object over its volume, on the other hand, will yield the larger value M 1 = ∫ 0 R 4 π r 2 ρ 1 − 2 G m r c 2 d r The difference between the

Kingston House, Shrewsbury

Kingston House is a building in St Alkmund's Place, Shrewsbury. It is a Grade II listed building; the house was built to a timber-frame design and completed in 1679. It has an unusual tower with a pyramid roof, it became a training facility for "friendless girls in moral danger" in 1872 and went on to become the headquarters of the Shropshire Regiment of Yeomanry Cavalry in the late 19th century. This unit evolved to become the Shropshire Imperial Yeomanry in 1901 and the Shropshire Yeomanry in 1908. By the early 20th century the Divisional Troops of the Royal Artillery and the offices of the Shropshire Territorial Force Association were based in the building; the Shropshire Yeomanry was mobilised at Kingston House in August 1914 before being deployed to Egypt. After the war the house was decommissioned and converted for commercial use: it is now occupied by a firm of solicitors