box

box, common name for the Buxaceae, a family of trees and shrubs with leathery evergreen leaves, native to the tropics and subtropics of the Old World and to Central America. The boxes (genus Buxus) have been widely introduced to other regions for use as hedge plants and for their wood. Boxwood is close-grained, strong and hard, and polishes well; it is valued for wood engraving, carving, and turning, and for making musical instruments. Pachysandra procumbens, a native American species of an otherwise Asian genus, is a low, creeping herb found in the S Appalachians and cultivated elsewhere as a ground cover. The box family is classified in the division Magnoliophyta, class Magnoliopsida, order Euphorbiales.

or voice box

Hollow, tubular structure connecting the pharynx with the trachea, through which air passes on the way to the lungs. The larynx consists of a framework of cartilage plates, with a ridge in front (Adam's apple); the epiglottis, a flaplike projection up into the throat that covers the airway during swallowing to keep food and liquid from entering; and the vocal cords, whose vibration produces the sound of the voice (see speech).

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Mechanical musical instrument in which projecting pins on a revolving brass cylinder or disk, encoding a piece of music, pluck tuned steel tongues. It was probably invented circa 1780 in Switzerland. With its modular cylinders or disks, it was a popular domestic instrument until displaced by the player piano and phonograph.

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Any of several species of terrestrial turtles found in the U.S. and Mexico (genus Terrapene) or the subtropical and tropical regions of Asia (genus Cuora). Box turtles have a high, rounded upper shell (carapace) that grows to a maximum length of about 7 in. (18 cm). The lower shell (plastron) is hinged across the centre and can be drawn very tightly against the carapace to form a protective “box” that completely encloses the turtle's soft parts. Box turtles feed on earthworms, insects, mushrooms, and berries and are often kept as pets.

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Hardy and fast-growing tree (Acer negundo), also called ash-leaved maple, of the maple family, native to the central and eastern U.S. Its compound leaves (rare among maples) consist of three, five, or seven coarsely toothed leaflets. The single seed is borne in a samara (dry, winged fruit). Because of its rapid growth and its drought resistance, it was widely planted for shade by early settlers in the prairie regions of the U.S. Maple syrup and sugar are sometimes obtained from the box elder. Its wood is used for crates, furniture, paper pulp, and charcoal.

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Instrument that records the performance and condition of an aircraft in flight. Regulatory agencies require these devices on commercial aircraft to make possible the analysis of crashes or other unusual occurrences. They are housed in heavy steel within layers of insulation, protecting them against impacts and fires. The recording tape is also protected against inadvertent erasure and contact with seawater. It records airspeed, altitude, heading, vertical acceleration, and aircraft pitch. It also includes a separate device that records voice communication within the aircraft and by radio. Both recorders are carried in the tail of the aircraft.

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A Box-Muller transform (by George Edward Pelham Box and Mervin Edgar Muller 1958) is a method of generating pairs of independent standard normally distributed (zero expectation, unit variance) random numbers, given a source of uniformly distributed random numbers.

It is commonly expressed in two forms. The basic form as given by Box and Muller takes two samples from the uniform distribution on the interval (0, 1] and maps them to two normally distributed samples. The polar form takes two samples from a different interval, [−1, +1], and maps them to two normally distributed samples without the use of sine or cosine functions.

One could use the inverse transform sampling method to generate normally-distributed random numbers instead; the Box-Muller transform was developed to be more computationally efficient. The more efficient Ziggurat algorithm can also be used.

Basic form

Suppose U₁ and U₂ are independent random variables that are uniformly distributed in the interval (0, 1]. Let

$Z\_0\; =\; R\; cos(Theta)\; =sqrt\{-2\; ln\; U\_1\}\; cos(2\; pi\; U\_2),$

and

$Z\_1\; =\; R\; sin(Theta)\; =\; sqrt\{-2\; ln\; U\_1\}\; sin(2\; pi\; U\_2).,$

Then Z₀ and Z₁ are independent random variables with a normal distribution of standard deviation 1.

The derivation is based on the fact that, in a two-dimensional cartesian system where X and Y coordinates are described by two independent and normally distributed random variables, the random variables for R² and Θ (shown above) in the corresponding polar coordinates are also independent and can be expressed as

$R^2\; =\; -2cdotln\; U\_1,$

and

$Theta\; =\; 2pi\; U\_2.,$

Polar form

The polar form is attributed by Devroye to Marsaglia. It is also mentioned without attribution in Carter.

Given u and v, independent and uniformly distributed in the closed interval [−1, +1], set s = R² = u² + v². (Clearly $scriptstyle\; R\; =\; sqrt\{s\}$.) If s = 0 or s > 1, throw u and v away and try another pair (u, v). Continue until a pair with s in the open interval (0, 1) is found. Because u and v are uniformly distributed and because only points within the unit circle have been admitted, the values of s will be uniformly distributed in the open interval (0, 1), too. The latter can be seen by calculating the cumulative distribution function for s in the interval (0, 1). This is the area of a circle with radius $scriptstyle\; sqrt\{s\}$ divided by $scriptstylepi$. From this we find the probability density function to have the constant value 1 on the interval (0, 1). Equally so, the angle θ divided by $scriptstyle\; 2\; pi$ is uniformly distributed in the open interval (0, 1) and independent of s.

We now identify the value of s with that of U₁ and $scriptstyle\; theta/(2\; pi)$ with that of U₂ in the basic form. As shown in the figure, the values of $scriptstyle\; cos\; theta\; =\; cos\; 2\; pi\; U\_2$ and $scriptstyle\; sin\; theta\; =\; sin\; 2\; pi\; U\_2$ in the basic form can be replaced with the ratios $scriptstylecos\; theta\; =\; u/R\; =\; u/sqrt\{s\}$ and $scriptstylesin\; theta\; =\; v/R\; =\; v/sqrt\{s\}$, respectively. The advantage is that calculating the trigonometric functions directly can be avoided. This is helpful when they are comparatively more expensive than the single division that replaces each one.

Just as the basic form produces two standard normal deviates, so does this alternate calculation.

$z\_0\; =\; sqrt\{-2\; ln\; U\_1\}\; cos(2\; pi\; U\_2)\; =\; sqrt\{-2\; ln\; s\}\; left(frac\{u\}\{sqrt\{s\}\}right)\; =\; u\; cdot\; sqrt\{frac\{-2\; ln\; s\}\{s\}\}$

and

$z\_1\; =\; sqrt\{-2\; ln\; U\_1\}\; sin(2\; pi\; U\_2)\; =\; sqrt\{-2\; ln\; s\}left(frac\{v\}\{sqrt\{s\}\}right)\; =\; v\; cdot\; sqrt\{frac\{-2\; ln\; s\}\{s\}\}.$

Contrasting the two forms

The polar method differs from the basic method in that it is a type of rejection sampling. It throws away some generated random numbers, but it is typically faster than the basic method because it is simpler to compute (provided that the random number generator is relatively fast) and is more numerically robust. It avoids the use of trigonometric functions, which are comparatively expensive in many computing environments. It throws away 1 − π/4 ≈ 21.46% of the total input uniformly distributed random number pairs generated, i.e. throws away 4/π − 1 ≈ 27.32% uniformly distributed random number pairs per Gaussian random number pair generated, requiring 4/π ≈ 1.2732 input random numbers per output random number.

The basic form requires three multiplications, one logarithm, one square root, and one trigonometric function for each normal variate.

The polar form requires two multiplications, one logarithm, one square root, and one division for each normal variate. The effect is to replace one multiplication and one trigonometric function with a single division.

See also

• Normal distribution
• Inverse transform sampling
• Marsaglia polar method, an optimization of the Box-Muller transform
• Ziggurat algorithm, a very different way to generate normal random numbers

References

External links

• Box-Muller running in a Java Applet
• Generating Gaussian Random Numbers
• Box-Muller Transform on MathWorld