It is assumed that the call arrivals can be modeled by a Poisson process and that call holding times are described by a negative exponential distribution.

Details for the required modifications to the test statistic and for the critical values for the normal distribution and the exponential distribution have been published, and later publications also include the Gumbel distribution.

Cho and Garcia-Molina show that the exponential distribution is a good fit for describing page changes, while Ipeirotis et al.

A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval ( 0, 1 ), the variate

has an exponential distribution, where F < sup > − 1 </ sup > is the quantile function, defined by

* The Benktander Weibull distribution reduces to a truncated exponential distribution

* The exponential distribution is a limit of a scaled beta distribution:

* Hyper-exponential distribution – the distribution whose density is a weighted sum of exponential densities.

* Hypoexponential distribution – the distribution of a general sum of exponential random variables.

* exGaussian distribution – the sum of an exponential distribution and a normal distribution.

The exponential distribution occurs naturally when describing the lengths of the inter-arrival times in a homogeneous Poisson process.

In contrast, the exponential distribution describes the time for a continuous process to change state.

But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p. m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives.

Fitted cumulative exponential distribution to annually maximum 1-day rainfalls using CumFreq

Reliability theory and reliability engineering also make extensive use of the exponential distribution.

Several elementary functions which are defined via power series may be defined in any unital Banach algebra ; examples include the exponential function and the trigonometric functions, and more generally any entire function.

The exponential function e < sup > x </ sup > for real values of x may be defined in a few different equivalent ways ( see Characterizations of the exponential function ).

Note that this derivation does assume that ƒ ( x ) is sufficiently differentiable and well-behaved ; specifically, that ƒ may be approximated by polynomials ; equivalently, that ƒ is a real analytic function of exponential type.

The exponential operator on the right hand side of the Schrödinger equation is usually defined by the corresponding power series in H. One might notice that taking polynomials or power series of unbounded operators that are not defined everywhere may not make mathematical sense.

) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently.

They may then be applied in pre-and post-multiplication to the quaternion representation of the coherency matrix, with the usual exploitation of the quaternion exponential for performing rotations and boosts taking a form equivalent to the matrix exponential equations above.

The accuracy of a predictive distribution may be measured using the distance or divergence between the true exponential distribution with rate parameter, λ < sub > 0 </ sub >, and the predictive distribution based on the sample x.

It provides a quadratic speedup, unlike other quantum algorithms, which may provide exponential speedup over their classical counterparts.

For example, a conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of θ.

If it is desired to sample more of the exponential electron portion of the characteristic, an asymmetric double probe may be used, with one electrode larger than the other.

Even when they terminate, parsers that use recursive descent with backup may require exponential time.

) The mathematical motivation for this type of notation, as well as additional Stirling number formulae, may be found on the page for Stirling numbers and exponential generating functions.

The response curve may be linear or exponential.

A conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of the angle indicating direction from the axis.

These problems are all hard: the clique decision problem is NP-complete ( one of Karp's 21 NP-complete problems ), the problem of finding the maximum clique is both fixed-parameter intractable and hard to approximate, and listing all maximal cliques may require exponential time as there exist graphs with exponentially many maximal cliques.

Although it has been believed that simple implementations of top-down parsing cannot accommodate direct and indirect left-recursion and may require exponential time and space complexity while parsing ambiguous context-free grammars, more sophisticated algorithms for top-down parsing have been created by Frost, Hafiz, and Callaghan which accommodate ambiguity and left recursion in polynomial time and which generate polynomial-size representations of the potentially exponential number of parse trees.

Exact discretization may sometimes be intractable due to the heavy matrix exponential and integral operations involved.

Simple implementations of top-down parsing do not terminate for left-recursive grammars, and top-down parsing with backtracking may have exponential time complexity with respect to the length of the input for ambiguous CFGs.

When top-down parser tries to parse an ambiguous input with respect to an ambiguous CFG, it may need exponential number of steps ( with respect to the length of the input ) to try all alternatives of the CFG in order to produce all possible parse trees, which eventually would require exponential memory space.

But the probability of survival of a new type may be quite low even if λ > 1 and the population as a whole is experiencing quite strong exponential growth | exponential increase.

They can be obtained analytically, meaning that the resulting orbitals are products of a polynomial series, and exponential and trigonometric functions.

Puzzle Bobble caters to this interest very well, featuring an exponential scoring system which allows extremely high scores to be achieved.

( In particular, the exponential map can be used to define abstract index groups.

In category theory, currying can be found in the universal property of an exponential object, which gives rise to the following adjunction in cartesian closed categories: There is a natural isomorphism between the morphisms from a binary product and the morphisms to an exponential object.

While the technique extends to higher dimension ( as proved by Edelsbrunner and Shah ), the runtime can be exponential in the dimension even if the final Delaunay triangulation is small.

The most widely used method of discounting is exponential discounting, which values future cash flows as " how much money would have to be invested currently, at a given rate of return, to yield the cash flow in future.

By starting with the field of rational functions, two special types of transcendental extensions ( the logarithm and the exponential ) can be added to the field building a tower containing elementary functions.

It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment.

( The trigonometric functions are in fact closely related to and can be defined via the exponential function using Euler's formula ).

The definition in general is somewhat technical, but in the case of real matrix groups, it can be formulated via the exponential map, or the matrix exponent.

The exponential function can be extended to a function which gives a complex number as e < sup > x </ sup > for any arbitrary complex number x ; simply use the infinite series with x complex.

This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm.

The first law was published in 1774 and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error, disregarding sign.

Furthermore, because of the exponential growth of the pot size in pot-limit play, seeing one of these hands to the end can be very expensive and carry immense reverse implied odds.

In particular, it can be seen that the Néel relaxation time is an exponential function of the grain volume, which explains why the flipping probability becomes rapidly negligible for bulk materials or large nanoparticles.

In Weaver's method, the band of interest is first translated to be centered at zero, conceptually by modulating a complex exponential with frequency in the middle of the voiceband, but implemented by a quadrature pair of sine and cosine modulators at that frequency ( e. g. 2 kHz ).

For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample ( i. e., a natural number n for which the Mertens function M ( n ) equals or exceeds the square root of n ) is known: all numbers less than 10 < sup > 14 </ sup > have the Mertens property, and the smallest number which does not have this property is only known to be less than the exponential of 1. 59 × 10 < sup > 40 </ sup >, which is approximately 10 to the power 4. 3 × 10 < sup > 39 </ sup >.