no. 2
Probability theory/Statistics
On the distribution of the product of correlated normal random variables
[Sur la distribution exacte du produit de variables aléatoires normales corrélées]

Présenté par : the Editorial Board

Nadarajah, Saralees1  ; Pogány, Tibor K.2
1 School of Mathematics, University of Manchester, Manchester M13 9PL, UK
2 Faculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Croatia
Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 201-204.

Dans cette Note, on résout un problème, posé depuis 1936, sur la distribution exacte du produit de variables aléatoires normales corrélées. Comme résultat supplémentaire, on déduit la distribution exacte de la moyenne du produit de variables aléatoires normales corrélées.

We solve a problem that has remained unsolved since 1936 – the exact distribution of the product of two correlated normal random variables. As a by-product, we derive the exact distribution of the mean of the product of correlated normal random variables.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.10.019
Nadarajah, Saralees 1 ; Pogány, Tibor K. 2

1 School of Mathematics, University of Manchester, Manchester M13 9PL, UK
2 Faculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Croatia 
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Nadarajah, Saralees; Pogány, Tibor K. On the distribution of the product of correlated normal random variables. Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 201-204. doi : 10.1016/j.crma.2015.10.019. http://www.numdam.org/articles/10.1016/j.crma.2015.10.019/

[1] Aroian, L.A. The probability function of the product of two normally distributed variables, Ann. Math. Stat., Volume 18 (1944), pp. 265-271

[2] Aroian, L.A.; Taneja, V.S.; Cornwell, L.W. Mathematical forms of the distribution of the product of two normal variables, Commun. Stat., Theory Methods, Volume 7 (1978), pp. 165-172

[3] Bandi, M.M.; Connaughton, C. Craig's XY distribution and the statistics of Lagrangian power in two-dimensional turbulence, Phys. Rev. E, Volume 77 (2008)

[4] Craig, C.C. On the frequency function of xy, Ann. Math. Stat., Volume 7 (1936), pp. 1-15

[5] Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products, Academic Press, San Diego, CA, USA, 2000

[6] Haldane, J.B.S. Moments of the distributions of powers and products of normal variates, Biometrika, Volume 32 (1942), pp. 226-242

[7] Hayya, J.C.; Ferrara, W.L. On normal approximations of the frequency functions of standard forms where the main variables are normally distributed, Manag. Sci., Volume 19 (1972), pp. 173-186

[8] MacKinnon, D. Introduction to Statistical Mediation Analysis, Routledge, New York, 2012

[9] MacKinnon, D.P.; Fritz, M.S.; Williams, J.; Lockwood, C.M. Distribution of the product confidence limits for the indirect effect: program PRODCLIN, Behav. Res. Methods, Volume 39 (2007), pp. 384-389

[10] Meeker, W.Q.; Odeh, R.D.; Cornwell, L.W.; Aroian, L.A.; Kennedy, W.J. Selected Tables in Mathematical Statistics: The Product of Two Normally Distributed Random Variables, American Mathematical Society, Providence, RI, USA, 1981

[11] Ware, R.; Lad, F. Approximating the distribution for sums of products of normal variables, Department of Mathematics and Statistics, University of Canterbury, New Zealand, 2013 (Working paper)

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