An Approximation to the Probability Normal

Ingeniería Investigación y Tecnología, volumen XVI (número 4), octubre-diciembre 2015: 605-611 ISSN 1405-7743 FI-UNAM (artículo arbitrado) doi: http://dx.doi.org/10.1016/j.riit.2015.09.012

An Approximation to the Probability Normal Distribution and its Inverse Una aproximación a la distribución de probabilidad normal y su inversa Alamilla-López Jorge Luis Instituto Mexicano del Petróleo Dirección de Investigación en Transformación de Hidrocarburos E-mail: [email protected]

Information on the article: received: February 2015, accepted: March 2015

Abstract Mathematical functions are used to compute Normal probabilities, which absolute errors are small; however their large relative errors make them unsuitable to compute structural failure probabilities or to compute the menace curves of natural hazards. In this work new mathematical functions are proposed to compute Normal probabilities and their inverses in an easy and accurate way. These functions are valid over a wide range of random varia‹•ŽȱŠ—ȱŠ›ŽȱžœŽž•ȱ’—ȱŠ™™•’ŒŠ’˜—œȱ ‘Ž›ŽȱŒ˜–™žŠ’˜—Š•ȱœ™ŽŽȱŠ—ȱŽĜŒ’Ž—Œ¢ȱ are required. In addition, these functions have the advantage that the numerical correspondence between the random value X = x and its Normal proba‹’•’¢ȱ̘ȱǻȬx) is bijective.

Keywords: ‡ ‡ ‡ ‡

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Resumen Se utilizan funciones matemáticas para calcular probabilidades normales, cuyos errores absolutos son pequeños; sin embargo, sus grandes errores relativos las hacen inadecuadas para calcular probabilidades de falla de estructuras o para calcular curvas de amenaza de peligros naturales. En este trabajo se proponen nuevas funciones matemáticas para calcular probabilidades normales y sus inversas, de manera fácil y precisa. Estas funciones son válidas en un intervalo amplio de valores que puede tomar la variable aleatoria y son útiles en aplicaciones donde se ›Žšž’Ž›ŽȱŸŽ•˜Œ’Šȱ¢ȱŽęŒ’Ž—Œ’ŠȱŒ˜–™žŠŒ’˜—Š•ǯȱŽ–¤œǰȱŽœŠœȱž—Œ’˜—Žœȱ’Ž—Ž—ȱ•Šȱ ventaja de que la correspondencia numérica entre el valor aleatorio X = x y su probabilidad Normal ̘ȱǻȬx) es biyectiva.

Descriptores: ‡ ‡ ‡ ‡

distribución normal inversa error absoluto error relativo

An Approximation to the Probability Normal Distribution and its Inverse

Introduction ‘Žȱ—˜›–Š•ȱ’œ›’‹ž’˜—ȱž—Œ’˜—ȱ̘ȱǻȬx) is used in engineering probabilistic applications to compute reliabili’Žœȱ ˜ȱ œ›žŒž›Š•ȱ œ¢œŽ–œȱ œžŒ‘ȱ Šœȱ ‹ž’•’—œǰȱ ˜ěœ‘˜›Žȱ platforms, pipelines, tanks, and bridges, among others. These applications require accurate estimations of small probabilities, which are associated with relatively large values. In addition, these probabilities have to be easily invertible. The disadvantage of not having a cloœŽȱŠ—Š•¢’ŒŠ•ȱ˜›–ȱ˜ȱŽœ’–Š’˜—ȱ̘ȱǻȬx) has been overcome using mathematical approximations. Summaries of these mathematical approximations are given in ǻ‹›Š–˜ ’ĵȱ ǭȱ Žž—ǰȱ ŗşŝŘDzȱ ŠŽ•ȱ ǭȱ ŽŠǰȱ ŗşşŜǼǰȱ which are useful to compute probabilities or their inverses, but not both. The errors of these approximations Š›Žȱœ™ŽŒ’ꮍȱ’—ȱŽ›–œȱ˜ȱ‘Ž’›ȱ–Š¡’–ž–ȱŠ‹œ˜•žŽȱŽ››˜›ǯȱ The absolute errors of the approximations are small but ‘Ž’›ȱ›Ž•Š’ŸŽȱŽ››˜›œȱŠ›Žȱœ’—’ęŒŠ—ǰȱ ‘’Œ‘ȱ‹ŽŒ˜–Žœȱ’–portant in the tail of probability distribution. In the present work, this inconvenience is shown for the best –Š‘Ž–Š’ŒŠ•ȱž—Œ’˜—ȱ›Ž™˜›Žȱ’—ȱǻ‹›Š–˜ ’ĵȱǭȱŽž—ǰȱŗşŝŘǼǰȱ‘Žȱ˜—Žȱ ’‘ȱ–’—’–ž–ȱŠ‹œ˜•žŽȱŽ››˜›ȱŠ–˜—ȱ ‘Žȱ ŠŸŠ’•Š‹•Žȱ Š™™›˜¡’–Š’˜—œǯȱ •œ˜ǰȱ ˜‘Ž›ȱ Š™™›˜¡’–Štions commonly used to compute failure probabilities are revised and new mathematical expressions with no such inconvenience are proposed. These new functions are useful to compute Normal probabilities and their inverses in a relatively easy and fast way, with good accuracy. Furthermore, they have the advantages of being valid over a wide range of the random variable, which makes them useful in engineering applications.

ž—Œ’˜—ȱΠȱǻx). It can also be seen as the one that transforms the probability density function in its cumulatiŸŽǯȱ Ž›Ž’—ǰȱ˜ȱ˜‹Š’—ȱŸŠ•žŽœȱ˜ȱΜǻȉǼȱ˜›ȱ•Š›ŽȱŸŠ•žŽœȱ˜ȱx, ‘Žȱ ꛜȱ Š—ȱ œŽŒ˜—ȱ Ž›’ŸŠ’ŸŽœȱ ΜǻȉǼȱ Š›Žȱ Š”Ž—DZȱ ΜȂǰȱ ΜȂȂ respectively, hence it gives the linear second order di쎛Ž—’Š•ȱŽšžŠ’˜— Μȱȁȁȱx Μȱȁȱ Μȱ ȱƽȱŖȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱǻŘǼ ‘’œȱ ’쎛Ž—’Š•ȱ ŽšžŠ’˜—ȱ  Šœȱ œ˜•ŸŽȱ —ž–Ž›’ŒŠ••¢ȱ ˜›ȱ ‘Žȱ’—’’Š•ȱŒ˜—’’˜—œȱΜ ǻx = 0) = ŘS / Ř Š—ȱΜȂȱǻx = 0) ƽȱȭŗǯȱžŒ‘ȱŠȱœ˜•ž’˜—ȱ’œȱŠ”Ž—ȱŠœȱ›ŽŽ›Ž—ŒŽȱ˜›ȱΜȱǻ˜).ȱœȱ œ‘˜ —ȱ ’—ȱ ’ž›Žȱ ŗǰȱ ‘Žȱ Šœ¢–™˜’Œȱ Š™™›˜¡’–Š’˜—ȱ ˜ȱ ŽœŒ›’‹ŽȱΜȱǻxǼǰȱ’ŸŽ—ȱ‹¢ȱ‘ŽȱŽ¡™›Žœœ’˜—ȱǻřǼǰȱ’ŸŽœȱ˜˜ȱ results for large x values. ଵ

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For xȱŸŠ•žŽœȱŒ•˜œŽȱ˜ȱ£Ž›˜ǰȱšǯȱǻřǼǰȱ’ŸŽ›Žœȱ›˜–ȱ‘ŽȱŒ˜››Žœ™˜—’—ȱ—ž–Ž›’ŒŠ•ȱž—Œ’˜—ǯȱ•‘˜ž‘ȱ‘ŽȱŠœ¢–™totic approximation of this function can be used to calculate probabilities associated with large values of x, it has the disadvantage of not being easily invertible.  ˜ȱž—Œ’˜—œȱ˜ȱŽœŒ›’‹ŽȱΜ ǻx), that provide good approximation over a wide range of x values, are those ™›˜™˜œŽȱ‹¢ȱ Šœ’—œȱǻ‹›Š–˜ ’ĵȱŠ—ȱŽž—ǰȱŗşŝŘDzȱ ǻ˜œŽ—‹•žŽ‘ǰȱŗşŞśǼǯȱ‘Žȱ Šœ’—œȂœȱŠ™™›˜ŠŒ‘ȱ’œȱŠȱŒ•Šœsic function, probably the most used to compute Normal probabilities, because it has the minimum absolute Ž››˜›ȱ’—ȱǻ‹›Š–˜ ’ĵȱŠ—ȱŽž—ǰȱŗşŝŘǼǰȱŠ—ȱŒŠ—ȱ‹ŽȱŽ¡pressed as follows

Reference framework Herein, some of the mathematical functions typically used in engineering applications are reviewed. The number of functions analyzed is not exhaustive; the author only discusses some approaches that in his opinion are representative for estimating probabilities and/or their inverses in structural and mechanical engineering. The analysis is focused on identifying the accuracy and scope of these functions, and showing the inconveniences mentioned in the previous section. In general, the normal probability distribution ž—Œ’˜—ȱ̘ȱǻ·) can be described in terms of its probabili¢ȱŽ—œ’¢ȱž—Œ’˜—ȱΠȱǻȉǼȱŠœȱ˜••˜ œ ̘ǻxǼȱƽȱΜǻxǼȱΠǻx)

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Ingeniería Investigación y Tecnología, volumen XVI (número 4), octubre-diciembre 2015: 605-611 ISSN 1405-7743 FI-UNAM

Alamilla-López Jorge Luis

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Ingeniería Investigación y Tecnología, volumen XVI (número 4), octubre-diciembre 2015: 605-611 ISSN 1405-7743 FI-UNAM

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An Approximation to the Probability Normal Distribution and its Inverse

interval [x0i , x0 i+ŗ], i = ǿŗǰȱŘǰȱřȀǰȱ ‘Ž›Žȱx0i and x0 iƸŗ values Žę—Žȱ‘Žȱ‹˜ž—Š›¢ȱ˜ȱŒ˜—’ž˜žœȱ’—Ž›ŸŠ•œǰȱœ™ŽŒ’ꮍȱ ’—ȱŠ‹•Žȱŗǯ ߰଴೔ ሺ‫ݔ‬ሻ ൌ ܿ଴೔ ݁‫݌ݔ‬ൣെ൫ܿଷ೔ ‫ ݔ‬ଷ ൅ ܿଶ೔ ‫ ݔ‬ଶ ൅ ܿଵ೔ ‫ݔ‬൯൧

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‘Žȱž—Œ’˜—ȱΜǻȉǼȱŽŒ›ŽŠœŽœȱŠœȱŠȱž—Œ’˜—ȱ˜ȱx. The expo—Ž—’Š•ȱž—Œ’˜—ȱ’ŸŽ—ȱ‹¢ȱšǯȱǻŜǼȱ’œȱœž’Š‹•Žȱ˜›ȱŽœŒ›’bing that decrement, because its behavior is decreasing. ‘Žȱ‘’›ȱŽ›ŽŽȱ™˜•¢—˜–’Š•ȱ ’‘ȱŒ˜—œŠ—ȱŒ˜ŽĜŒ’Ž—œȱ cŗi, cŘi, cři present in the argument of the exponential function, was chosen due to its simplicity, since it allows the roots of a third degree polynomial be evaluated in a closed way. Besides, the natural logarithm of šǯȱǻŜǼȱ’œȱŠ•œ˜ȱŠȱ™˜•¢—˜–’Š•ȱ˜ȱ‘’›ȱŽ›ŽŽȱŠ—ȱ’—ŸŽ›’ble. It is noteworthy that the resulting polynomial of šǯȱǻŗǼȱŒŠ——˜ȱ‹Žȱ•Žœœȱ‘Š—ȱ ˜ǰȱ‹ŽŒŠžœŽȱ‘ŽȱŠ›ž–Ž—ȱ˜ȱȱ ΠǻȉǼȱ’œȱŠȱ™˜•¢—˜–’Š•ȱ˜ȱ›ŠŽȱ ˜ǯȱȱ‘’›ȱŽ›ŽŽȱ™˜•¢—˜–’Š•ȱ‘Šœȱ‘ŽȱŠŸŠ—ŠŽȱ˜ȱŠ••˜ ’—ȱ‹ŽĴŽ›ȱ›Ž™›ŽœŽ—Š’˜—ȱ˜ȱ‘Žȱž—Œ’˜—ȱΜǻȉǼǰȱ ’‘ȱŠȱœ–Š••Ž›ȱ—ž–‹Ž›ȱ˜ȱŒ˜—Ȭ tiguous intervals. ȱ’œȱŠœœž–Žȱ‘ŠȱΜȱǻȉǼȱ’œȱŽšžŠ•ȱ˜ȱ‘ŽȱŒ˜››Žœ™˜—’—ȱ ž—Œ’˜—ȱΜ0i ǻȉǼǰȱi = ǿŗǰȱŘǰȱřȀǰȱȱ’—ȱ‘Žȱ’‘ȱœŽ–Ž—ǯȱ‘ŽȱŸŠ•žŽœȱ ˜ȱ Œ˜ŽĜŒ’Ž—œȱ cŘi and cři ǰȱ ˜‹Š’—Žȱ ›˜–ȱ Šȱ ęȱ ˜›ȱ each interval [x0i , x0 i+1 ǾǰȱŠ›Žȱœ‘˜ —ȱ’—ȱŠ‹•ŽȱŗDzȱ–ŽŠ— ‘’•Žȱ‘ŽȱŒ˜ŽĜŒ’Ž—œȱc0i and cŗi ǰȱŠ›Žȱœ™ŽŒ’ꮍȱ’—ȱŽ›–œȱ˜ȱcŘi and cři , by means of the following expressions

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‫ݎ‬ଷ೔ ൌ െ‫ݎ‬଴೔ ൫‫ݔ‬଴ଷ೔ െ ‫ݔ‬଴ଷ೔శభ ൯ȱ

ǻŗŖǯǼ

\ i \ǻ x0 Ǽ and \ i ŗ \ǻ xŖ Ǽ ŸŠ•žŽœȱŠ›Žȱœ™ŽŒ’ꮍȱ’—ȱŠ‹•Žȱ x0 and x0 vaŗǰȱ ‘’Œ‘ȱŠ›Žȱ›Ž•ŠŽȱ˜ȱ‘ŽȱŒ˜››Žœ™˜—’—ȱȱȱȱ •žŽœǯȱ˜›ȱ‘ŽȱꛜȱœŽ–Ž—ǰȱ‘ŽȱŸŠ•žŽœȱ˜ȱcŖŗȱȱŠ—ȱΜŗ can S Ȧ Ř . ‘ŽȱŸŠ•žŽœȱ˜ȱŒ˜ŽĜbe also obtained as cŖ \ŗ Œ’Ž—œȱ’—ȱŠ‹•ŽȱŗȱŠ—ȱ‘Žȱ›Ž•Š’˜—œȱŝȱ˜ȱŗŖȱœŠ’œ¢ȱ‘Šȱ‘Žȱ values estimates of the distribution function be conti—ž˜žœǯȱ œȱ œ‘˜ —ȱ ’—ȱ ’ž›Žȱ śǰȱ ‘Žȱ ›Ž•Š’ŸŽȱ Ž››˜›œȱ Š›Žȱ •˜ Ž›ȱ‘Š—ȱȩΉRǻxǼȱǀȱŘǯśȱu_, which means that are be•˜ ȱŖǯŘśƖǯȱ‘Žȱ›Ž•Š’ŸŽȱŽ››˜›œȱŠœœ˜Œ’ŠŽȱ˜ȱx values of ŖǰȱřǰȱŞȱŠ—ȱŘŖȱŠ›ŽȱŽšžŠ•ȱ˜ȱ£Ž›˜ǰȱ ‘’Œ‘ȱ’œȱŒ˜—œ’œŽ—ȱ ’‘ȱ the proposed approach. The error is greater than that ˜‹Š’—Žȱ‹¢ȱǻ˜œŽ—‹•žŽ‘ǰȱŗşŞśǼȱ˜›ȱ˜—Žȱ˜›Ž›ȱ˜ȱ–Š—’žŽǯȱ ‘Žȱ Š‹œ˜•žŽȱ Ž››˜›ȱ ’œȱ •˜ Ž›ȱ ‘Š—ȱ ȩΉǻxǼȱ ǀȱ şǯśȱ u for x < řǰȱŠ—ȱ’œȱ•˜ Ž›ȱ‘Š—ȱȩΉǻxǼȱǀȱŜǯŚȱu_ for x tǯȱ’ž›ŽȱŜǰȱœ‘˜ œȱ‘Šȱ‘Žȱ™›˜™˜œŽȱŠ™™›˜ŠŒ‘ȱ’œȱ˜˜ȱ Ž—˜ž‘ȱ˜ȱŠŽšžŠŽ•¢ȱŽœŒ›’‹Žȱ‘Žȱž—Œ’˜—ȱΜǻȉǼǯȱŽœ’des, as is shown in the next section, the set of functions ’—ȱšǯȱǻŜǼȱ‘Šœȱ‘Žȱ›ŽŠȱŠŸŠ—ŠŽȱ˜ȱ‹Ž’—ȱ’—ŸŽ›’‹•Žǯ i ŗ

i

i ŗ

i

ŗ

0.003

ܿ଴೔ ൌ ݁‫݌ݔ‬ൣ‫ݍ‬ଵ೔ ൅ ‫ݍ‬ଶ೔ ܿଶ೔ ൅ ‫ݍ‬ଷ೔ ܿଷ೔ ൧ȱ ܿଵ೔ ൌ ‫ݎ‬ଵ೔ ൅ ‫ݎ‬ଶ ܿଶ೔ ൅ ‫ݎ‬ଷ೔ ܿଷ೔ ȱ

ǻŞǼ

௜

Where qki and rki ǰ i ǿŗǰ Řǰ řȀǰ k ǿŗǰ Řǰ řȀ , are given so ‘Šȱ‘ŽȱŸŠ•žŽœȱ˜ȱΜ0i ǻȉǼȱ’—ȱŒ˜—’ž˜žœȱ’—Ž›ŸŠ•œȱŠ›ŽȱŽšžŠ•ȱ ˜ȱΜǻȉǼǯȱ‘ŽœŽȱŒ˜ŽĜŒ’Ž—œȱŠ›Žȱ˜‹Š’—ŽȱŠœȱ˜••˜ œ ȱ‫ݍ‬ଵ೔ ൌ ‫ݎ‬ଵ೔ ‫ݔ‬ ȱ ଴೔ ൅ Ž ɗ௜ ȱ

ǻşǯŠǼ

‫ݍ‬ଶ೔ ൌ ‫ݎ‬ଶ೔ ‫ݔ‬଴೔ ൅  ‫ݔ‬଴ଶ೔ ȱ

ǻşǯ‹Ǽ

‫ݍ‬ଷ೔ ൌ ‫ݎ‬ଷ೔ ‫ݔ‬଴೔ ൅  ‫ݔ‬଴ଷ೔ ȱ

ǻşǯŒǼ

Relative error (in numerical value)

ǻŝǼ 0.002

0.001

0.000

-0.001

-0.002

-0.003

0

2

4

6

8

10

12

14

16

Š›Š–ŽŽ›œ

˜ŽĜŒ’Ž—œ

i

x0i

x0i+ŗ

Μi

Μi+ŗ

cŘi

cři

ŗ

0

ř

ŗǯŘśřřŗŚ

ŖǯřŖŚśş

ŖǯŖŗŚŞŚŚŞŜś

ƺŖǯŗŚŝŞŗŘřŘş

Ř

ř

Ş

ŖǯřŖŚśş

ŖǯŗŘřŗřŘ

ŖǯŖŖŗśşŝřŗř

ƺŖǯŖŚŗśŘŗŚşş

ř

Ş

ŘŖ

ŖǯŗŘřŗřŘ

ŖǯŖŚşŞŝŜ

ŖǯŖŖŖŗřŘŗŜŚ

ƺŖǯŖŖŞřŖŗřŘŝ

608

20

)LJXUH5HODWLYHHUURUDVDIXQFWLRQRIxRIWKHSURSRVHG DSSURDFK

7DEOH3DUDPHWHUVDQGFRHIILFLHQWVYDOXHVRIWKHSURSRVHGDSSURDFK Segment

18

x

Ingeniería Investigación y Tecnología, volumen XVI (número 4), octubre-diciembre 2015: 605-611 ISSN 1405-7743 FI-UNAM

Alamilla-López Jorge Luis

1

\(x)

Proposed approach Numerical

0.1

0

2

4

6

8

10

12

14

16

18

20

x

)LJXUH1XPHULFDODQGWKHSURSRVHGDSSURDFKRIIXQFWLRQ

Μȱǻ˜)

Input data: compute probability p for a given x value

Read values:  ‫ݔ‬଴೔ ǡ ‫ݔ‬଴೔శభ ǡ ߰௜ǡ ߰௜ାଵ ǡ ܿଶ೔ ǡ ܿଷ೔ , ݅ ൌ ሼͳǡ ǥ ǡ ͵ሽ, (Table 1)

Compute: Coefficients ‫ݎ‬௝೔ , ݆ ൌ ሼͲǡ ǥ ǡ͵ሽǡ ݅ ൌ ሼͳǡ ǥ ǡ ͵ሽ (Eqs. 10) Coefficients ‫ݍ‬௝೔ , ݆ ൌ ሼͳǡ ǥ ǡ͵ሽǡ ݅ ൌ ሼͳǡ ǥ ǡ ͵ሽ (Eqs. 9) Coefficients ܿ௝೔ , ݆ ൌ ሼͲǡͳሽǡ ݅ ൌ ሼͳǡ ǥ ǡ ͵ሽ (Eqs. 7-8)

Locate ȁ‫ݔ‬ȁ value in some interval: ሾ‫ݔ‬଴೔ ǡ ‫ݔ‬଴೔శభ ሻ for ݅ ൌ ሼͳǡ ǥ ǡ͵ሽ

Compute: ‫( ݔ‬Eq.6)

‫ݔ‬൑Ͳ

no

Compute: ‫ ݌‬ൌ ͳ െ ߰ሺ‫ݔ‬ሻ ߮ሺ‫ݔ‬ሻ (Eq. 1)

yes Compute: ‫ ݌‬ൌ ߰ሺȁ‫ݔ‬ȁሻ߮ሺ‫ݔ‬ሻ (Eq. 1)

)LJXUH)ORZFKDUWIRUFRPSXWLQJ 1RUPDOSUREDELOLWLHV

Ingeniería Investigación y Tecnología, volumen XVI (número 4), octubre-diciembre 2015: 605-611 ISSN 1405-7743 FI-UNAM

609

An Approximation to the Probability Normal Distribution and its Inverse

The parameters dji, j = ǿŖǰȱŗǰȱŘǰȱřȀǰȱ˜ȱŽŠŒ‘ȱœŽ–Ž—ȱi are ˜‹Š’—Žȱ’—ȱŽ›–œȱ˜ȱ‘ŽȱŒ˜ŽĜŒ’Ž—œȱcїi , j = ǿŖǰȱŗǰȱŘǰȱřȀȱ˜ȱ the corresponding segment i, by means of the expressions

Input data: compute x value for a given probability p

Read values: ‫ݔ‬଴೔ ǡ ‫ݔ‬଴೔శభ ǡ ߰௜ǡ ߰௜ାଵ ǡ ܿଶ೔ ǡ ܿଷ೔ , ݅ ൌ ሼͳǡ ǥ ǡ ͵ሽ, (Table 1)

݀଴೔ ൌ Compute: Coefficients ‫ݎ‬௝೔ , ݆ ൌ ሼͲǡ ǥ ǡ͵ሽǡ ݅ ൌ ሼͳǡ ǥ ǡ ͵ሽ (Eqs. 10) Coefficients ‫ݍ‬௝೔ , ݆ ൌ ሼͳǡ ǥ ǡ͵ሽǡ ݅ ൌ ሼͳǡ ǥ ǡ ͵ሽ (Eqs. 9) Coefficients ܿ௝೔ , ݆ ൌ ሼͲǡͳሽǡ ݅ ൌ ሼͳǡ ǥ ǡ ͵ሽ (Eqs. 7-8) Coefficients ݀௝೔ , ݆ ൌ ሼͲǡ ǥ ǡ͵ሽǡ ݅ ൌ ሼͳǡ ǥ ǡ ͵ሽ (Eqs. 13)

݀ଵ೔ ൌ

Locate Ȱሺ‫ݔ‬଴೔ ሻ ൑ ‫ ݌‬൏ Ȱሺ‫ݔ‬଴೔శభ ሻ value in some interval: ሾ‫ݔ‬଴೔ ǡ ‫ݔ‬଴೔శభ ሻ for ݅ ൌ ሼͳǡ ǥ ǡ͵ሽ, (flow diagram 1)

Compute: y (Eq.12)

no

‫ ݌‬൑ ͳΤʹ

‫ݔ‬ൌ‫ݕ‬

yes ‫ ݔ‬ൌ െ‫ݕ‬

)LJXUH)ORZFKDUWIRUFRPSXWLQJWKHLQYHUVHRIWKHQRUPDO GLVWULEXWLRQ

‘ŽȱŽ—Ž›’Œȱ̘ ȱŒ‘Š›ȱœ‘˜ —ȱ’—ȱ’ž›ŽȱŝȱŒŠ—ȱ‹Žȱ’–™•Žmented in a computer program. The purpose is to describe the steps to compute a Normal probability value for a given value of the variable. If a number of compu’—ȱ’œȱ—ŽŒŽœœŠ›¢ǰȱ‘ŽȱŒ˜ŽĜŒ’Ž—œȱ—ŽŽȱ—˜ȱž›‘Ž›ȱŽŸŠluation.

Inverses of normal probabilities ŒŒ˜›’—ȱ˜ȱ‘Žȱ•ŠœȱœŽŒ’˜—ǰȱž—Œ’˜—ȱΜȱǻȉǼȱŒŠ—ȱ‹ŽȱŽœcribed in three contiguous segments by means of ž—Œ’˜—œȱΜ0i ǻȉǼǰȱi = ǿŗǰȱŘǰȱřȀǰȱ’ŸŽ—ȱ‹¢ȱšǯȱǻŜǼǯȱ‘Žȱ’—ŸŽ›œŽȱ ˜ȱšǯȱǻŗǼȱŒŠ—ȱ‹ŽȱŽ¡™›ŽœœŽȱŠœ ‫ ݔ‬ൌ െȰିଵ ሾɗሺ‫ݔ‬ሻɔሺ‫ݔ‬ሻሿȱ

ǻŗŗǼ

and it can be computed as follows ‫ݔ‬ൌ

ௗబ೔ ଷ

ଵ

ௗమ೔ ିௗయ೔ ୪୬ ஍

ଷ

ටିௗభయ೔

െ ʹඥെ݀ଵ೔ ܿ‫ ݏ݋‬൮ ܿ‫ି ݏ݋‬ଵ ቌ

Ȱሺ‫ݔ‬଴೔ ሻ ൑ Ȱሺ‫ݔ‬ሻ ൏ Ȱሺ‫ݔ‬଴೔శభ ሻȱ

610

ቍ൲͕

ǻŗŘǼ

ͳ ͳ ൬ ൅ ܿଶ೔ ൰ȱ ܿଷ೔ ʹ ௖భ೔ ଷ௖య೔

െ

ௗబమ

೔

ଽ

ȱ

ȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱǻŗřǯŠǼ

ȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱǻŗřǯ‹Ǽ

݀ଶ೔ ൌ

݀଴ଷ೔ ͳ ͳ ቈͻܿଵ೔ ݀଴೔ െ ʹȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱǻŗřǯŒǼ െ ʹ͹ ൬ Ž ʹߨ െ Ž ܿ଴೔ ൰቉ȱ ͷͶܿଷ೔ ʹ ܿଷ೔

݀ଷ೔ ൌ

ʹ͹ ͳ  ȱ ͷͶ ܿଷ೔

ȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱǻŗřǯǼ

’ŸŽ—ȱ‘Šȱ‘ŽȱŒ˜ŽĜŒ’Ž—œȱdji and cji , j = ǿŖǰȱŗǰȱŘǰȱřȀȱŠ›Žȱ •’—”Žȱ‹¢ȱ–ŽŠ—œȱ˜ȱŽ¡™›Žœœ’˜—ȱŗŗǰȱ‘Žȱx values corres™˜—ȱ˜ȱŠȱœ™ŽŒ’ęŒȱ)x , which means that the numerical correspondences between these values is one to one ǻ‹’“ŽŒ’ŸŽǼǯ ‘ŽȱŽ—Ž›’Œȱ̘ ȱŒ‘Š›ȱœ‘˜ —ȱ’—ȱ’ž›ŽȱŞȱŒŠ—ȱ‹Žȱ’–plemented in a computer program. The purpose is to describe the steps to compute the inverse normal value for a given probability. This algorithm can be used alŽ›—Š’ŸŽ•¢ȱ˜ȱ‘ŽȱŠ•˜›’‘–œȱŽœŒ›’‹Žȱ‹¢ȱǻ —žǰȱŗşşŝǼȱ to simulate values of the Normal distribution. If a num‹Ž›ȱ˜ȱŒŠ•Œž•Š’˜—ȱ’œȱ—ŽŒŽœœŠ›¢ǰȱ‘ŽȱŒ˜ŽĜŒ’Ž—œȱ—˜ȱ—ŽŽȱ further evaluation.

Conclusions This work has proposed simple expressions that allow computing normal probabilities, even in the tail of the distribution, and their corresponding inverses. Such expressions are impressively manageable and useful for engineering applications. The proposed approach is simple because the resulting mathematical expressions are simple too, and its relative errors of ‘Žȱ —Ž ȱ Š™™›˜¡’–Š’˜—ȱ Š›Žȱ •˜ Ž›ȱ ‘Š—ȱ ȩΉRǻxǼȱ ǀȱ Řǯśȱ uŗŖ_ over a wide range of the random variable. The Š‹œ˜•žŽȱŽ››˜›ȱ’œȱ•˜ Ž›ȱ‘Š—ȱȩΉǻxǼȱǀȱşǯśȱuŗŖƺŚ_ for x ǀȱřǰȱ Š—ȱ’œȱ•˜ Ž›ȱ‘Š—ȱȩΉǻxǼȱǀȱŜǯŚȱuŗŖ_ for x tȱřǯȱ‘ŽȱŠ™proach does not have the inconveniences of typical approximations used in several applications that lead to relatively small absolute errors but significant relative errors, which become important in the tail of the probability distribution and are difficult ˜ȱ ’—ŸŽ›ǯȱ Ž•Š’ŸŽȱ Ž››˜›œȱ Š›Žȱ Œ›’’ŒŠ•ȱ  ‘Ž—ȱ ŠŒŒž›ŠŽȱ estimations of small probabilities are required, so

Ingeniería Investigación y Tecnología, volumen XVI (número 4), octubre-diciembre 2015: 605-611 ISSN 1405-7743 FI-UNAM

Alamilla-López Jorge Luis

this new approximation is a powerful tool to compute probabilities or/and their inverses in engineering applications.

References ‹›Š–˜ ’ĵȱǯȱŠ—ȱŽž—ȱǯǯȱHandbook of mathematical functions, ŗŖ‘ȱŽǯǰȱŽ ȱ˜›”ǰȱ˜ŸŽ›ȱž‹•’ŒŠ’˜—œǰȱŗşŝŘǰȱ™™ǯȱşŘśȬşŝŜǯ

—ž‘ȱǯǯȱThe art of computer programming, Vol. 2, Seminumerical algorithmsǰȱř‘ȱŽǯǰȱ’œ˜—ȬŽœ•Ž¢ǰȱŗşşŝǰȱ™™ǯȱŗȬȱŝŜŚǯ ›Š£ȱǯȱȱœ’–™•ŽȱŠ™™›˜¡’–Š’˜—ȱ˜ȱ‘Žȱ Šžœœ’Š—ȱ’œ›’‹ž’˜—ǯȱ Structural SafetyǰȱŸ˜•ž–ŽȱşǰȱŗşşŗDZȱřŗśȬřŗŞǯ ŠŽ•ȱ ǯ ǯȱŠ—ȱŽŠȱǯǯȱHandbook of the normal distribution, Statis’Œœȱ Ž¡‹˜˜”œȱ Š—ȱ ˜—˜›Š™‘œǰȱ ؗȱ Žǯǰȱ Ž ȱ ˜›”ǰȱ ȱ ›ŽœœǰȱŗşşŜǰȱ™™ǯȱŗȬŚŘŝǯ ˜œŽ—‹•žŽ‘ȱǯȱ—ȱŒ˜–™ž’—ȱ˜›–Š•ȱ›Ž•’Š‹’•’’ŽœǯȱStructural SafetyǰȱŸ˜•ž–ŽȱŘǰȱŗşŞśDZȱŗŜśȬŗŜŝǯ ŽŽ›ȱ ǯǯȱ—ȱ’—ŸŽ›’‹•ŽȱŠ™™›˜¡’–Š’˜—ȱ˜ȱ‘Žȱ—˜›–Š•ȱ’œ›’‹žtion function. ˜–™žŠ’˜—Š•ȱŠ’œ’ŒœȱǭȱŠŠȱ—Š•¢œ’œ, volume ŗŜǰȱŗşşřDZȱŗŗşȬŗŘřǯ

Citation for this article: Chicago style citation $ODPLOOD/ySH] -RUJH /XLV $Q DSSUR[LPDWLRQ WR WKH SUREDELOLW\ QRUPDOGLVWULEXWLRQDQGLWVLQYHUVHIngeniería Investigaciyn y Tecnología;9,  ISO 690 citation style $ODPLOOD/ySH] -/ $Q DSSUR[LPDWLRQ WR WKH SUREDELOLW\ QRUPDO GLVWULEXWLRQDQGLWVLQYHUVHIngeniería Investigaciyn y Tecnología YROXPHQ;9,Q~PHUR RFWXEUHGLFLHPEUH

About the author •Š–’••ŠȬà™Ž£ȱ ˜›Žȱž’œǯ The author is a civil engineer with master and doctorate de›ŽŽȱ’—ȱŽ—’—ŽŽ›’—ȱ›˜–ȱǯȱ Žȱ’œȱŠȱž••Ȭ’–Žȱ›ŽœŽŠ›Œ‘Ž›ȱŠȱ‘ŽȱŽ¡’ŒŠ—ȱŽ›˜leum Institute and teacher of postgraduate course in civil engineering at the School ˜ȱ—’—ŽŽ›’—ȱŠ—ȱ›Œ‘’ŽŒž›Žȱ˜ȱ‘ŽȱŠ’˜—Š•ȱ˜•¢ŽŒ‘—’Œȱ—œ’žŽǯȱ Žȱ’œȱŠȱŽŸŽ•ȱŘȱ’—ȱȱǻŠ’˜—Š•ȱ¢œŽ–ȱ˜ȱŽœŽŠ›Œ‘Ž›œǼǯȱȱ ’œȱ–Š’—ȱŠ›ŽŠȱ˜ȱ’—Ž›Žœȱ’œȱ˜ŒžœŽȱ on the modeling and assessment of the uncertain mechanical behavior of structural systems, on the stochastic modeling of loads and natural degrading phenomena, natural hazards, and on decision-making risk analysis. He published more than twenty papers in refereed journals.

Ingeniería Investigación y Tecnología, volumen XVI (número 4), octubre-diciembre 2015: 605-611 ISSN 1405-7743 FI-UNAM

611