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Expressions for the normal distribution and repeated normal integrals

Author

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  • Withers, C.S.
  • McGavin, P.N.

Abstract

We give a new expression for Mills ratio and five new expressions for repeated integrals of the univariate normal density, or equivalently for the Hermite functions. The Hermite functions are shown to be the negative moments of x+iN where N is a unit normal random variable and . This extends an earlier result that the Hermite polynomials are the positive moments of x+iN. We also give the derivatives of Mills' ratio and its inverse.

Suggested Citation

  • Withers, C.S. & McGavin, P.N., 2006. "Expressions for the normal distribution and repeated normal integrals," Statistics & Probability Letters, Elsevier, vol. 76(5), pages 479-487, March.
    • Handle: RePEc:eee:stapro:v:76:y:2006:i:5:p:479-487
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    References listed on IDEAS

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    1. Withers, C. S., 2000. "A simple expression for the multivariate Hermite polynomials," Statistics & Probability Letters, Elsevier, vol. 47(2), pages 165-169, April.
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    Cited by:

    1. Mwabu, Germano, 2008. "The Production of Child Health in Kenya: A Structural Model of Birth Weight," Working Papers 52, Yale University, Department of Economics.
    2. Christopher S. Withers & Saralees Nadarajah, 2011. "New Expressions for Repeated Upper Tail Integrals of the Normal Distribution," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 855-871, December.
    3. Germano Mwabu, 2008. "The Production of Child Health in Kenya: A Structural Model of Birth Weight," Working Papers 963, Economic Growth Center, Yale University.
    4. De Schrijver, Steven K. & Aghezzaf, El-Houssaine & Vanmaele, Hendrik, 2014. "Double precision rational approximation algorithm for the inverse standard normal second order loss function," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 247-253.
    5. Sun, Ping, 2007. "Moment representation of Bernoulli polynomial, Euler polynomial and Gegenbauer polynomials," Statistics & Probability Letters, Elsevier, vol. 77(7), pages 748-751, April.

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