Integral of the Bivariate Normal Distribution over an Offset Circle

Gilliland, Dennis

doi:10.1080/01621459.1962.10500813

Cited by 23 publications

(19 citation statements)

References 10 publications

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“…The truncation error bound by Groenewoud et [3] can be obtained by specializing the series (1) with p = a h , n=2. Since series (2) and (3) are mixtures, there exist simple upper bounds on their truncation errors.…”

Section: --mentioning

confidence: 99%

A note on some series representations of the integral of a bivariate normal distribution over an offset circle

Gilliland¹,

Hansen²

1974

Naval Research Logistics

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Consider the problem of computing an offset circle probability under a normal distribution. One approach is to utilize an infinite series representation in which case it is important to have rapid convergence and a good upper bound on the error introduced by consideration of only a finite number of terms of the series. We relate three seemingly different series representations. In particular we show how two series representations for the bivariate case can be obtained by specializing more general results of Harold Ruben.

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Section: --mentioning

confidence: 99%

A note on some series representations of the integral of a bivariate normal distribution over an offset circle

Gilliland¹,

Hansen²

1974

Naval Research Logistics

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“…We point the interested reader to Ref. [32] as a guide to the general form of the calculation. Instead, we numerically evaluateP n by stochastically sampling W |β,η (α) according to the prescription of Ref.…”

Section: B Squeezed Coherent Statementioning

confidence: 99%

Approximate particle number distribution from direct stochastic sampling of the Wigner function

2016

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We consider the Wigner quasi-probability distribution function of a single mode of an electromagnetic or matter-wave field to address the question of whether a direct stochastic sampling and binning of the absolute square of the complex field amplitude can yield a distribution functionPn that closely approximates the true particle number probability distribution Pn of the underlying quantum state. By providing an operational definition of the binned distributionPn in terms of the Wigner function, we explicitly calculate the overlap betweeñ Pn and Pn and hence quantify the statistical distance between the two distributions. We find that there is indeed a close quantitative correspondence betweenPn and Pn for a wide range of quantum states that have smooth and broad Wigner function relative to the scale of oscillations of the Wigner function for the relevant Fock state. However, we also find counterexamples, including states with high mode occupation, for whichPn does not closely approximate Pn.

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“…Gilliland (1962) suggests approximating P (R/u,ro! A) by the first few terms of the infinite A similar (but more accessible) approximation to P(R/c,r 0 !a) is given on p. 940 of Abramowitz and Stegun (1964): …”

Section: '?mentioning

confidence: 99%

“…However, much better approximations (not dependent upon c being near unity) are available. Gilliland (1962) suggests approximating P(R ) by the first few terms of the infinite series…”

Section: '?mentioning

confidence: 99%

Mathematical Models of Target Coverage and Missile Allocation

Eckler¹,

Burr²

1972

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Integral of the Bivariate Normal Distribution over an Offset Circle

Cited by 23 publications

References 10 publications

A note on some series representations of the integral of a bivariate normal distribution over an offset circle

A note on some series representations of the integral of a bivariate normal distribution over an offset circle

Approximate particle number distribution from direct stochastic sampling of the Wigner function

Mathematical Models of Target Coverage and Missile Allocation

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