# An Approach to Distribution of the Product of Two Normal Variables

• Volume: 32, Issue: 1-2, page 87-99
• ISSN: 1509-9423

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## Abstract

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The distribution of product of two normally distributed variables come from the first part of the XX Century. First works about this issue were [1] and [2] showed that under certain conditions the product could be considered as a normally distributed. A more recent approach is [3] that studied approximation to density function of the product using three methods: numerical integration, Monte Carlo simulation and analytical approximation to the result using the normal distribution. They showed as the inverse variation coefficient μ/σ increases, the distribution of the product of two independent normal variables tends towards a normal distribution. Our study is focused in Ware and Lad approaches. The objective was studying which factors have more influence in the presence of normality for the product of two independent normal variables. We have considered two factors: the inverse of the variation coefficient value μ/σ and the combined ratio (product of the two means divided by standard deviation): μ₁μ₂/σ for two normal variables with the same variance. Our results showed that for low values of the inverse of the variation coefficient (less than 1) normal distribution is not a good approximation for the product. Another one, influence of the combined ratio value is less than influence of the inverse of coefficients of variation value.

## How to cite

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Antonio Seijas-Macías, and Amílcar Oliveira. "An Approach to Distribution of the Product of Two Normal Variables." Discussiones Mathematicae Probability and Statistics 32.1-2 (2012): 87-99. <http://eudml.org/doc/271027>.

@article{AntonioSeijas2012,
abstract = { The distribution of product of two normally distributed variables come from the first part of the XX Century. First works about this issue were [1] and [2] showed that under certain conditions the product could be considered as a normally distributed. A more recent approach is [3] that studied approximation to density function of the product using three methods: numerical integration, Monte Carlo simulation and analytical approximation to the result using the normal distribution. They showed as the inverse variation coefficient μ/σ increases, the distribution of the product of two independent normal variables tends towards a normal distribution. Our study is focused in Ware and Lad approaches. The objective was studying which factors have more influence in the presence of normality for the product of two independent normal variables. We have considered two factors: the inverse of the variation coefficient value μ/σ and the combined ratio (product of the two means divided by standard deviation): μ₁μ₂/σ for two normal variables with the same variance. Our results showed that for low values of the inverse of the variation coefficient (less than 1) normal distribution is not a good approximation for the product. Another one, influence of the combined ratio value is less than influence of the inverse of coefficients of variation value. },
author = {Antonio Seijas-Macías, Amílcar Oliveira},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {product of normally distributed variables; inverse coefficient of variation; numerical integration; Monte Carlo simulation; combined ratio; normal distribution},
language = {eng},
number = {1-2},
pages = {87-99},
title = {An Approach to Distribution of the Product of Two Normal Variables},
url = {http://eudml.org/doc/271027},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Antonio Seijas-Macías
AU - Amílcar Oliveira
TI - An Approach to Distribution of the Product of Two Normal Variables
JO - Discussiones Mathematicae Probability and Statistics
PY - 2012
VL - 32
IS - 1-2
SP - 87
EP - 99
AB - The distribution of product of two normally distributed variables come from the first part of the XX Century. First works about this issue were [1] and [2] showed that under certain conditions the product could be considered as a normally distributed. A more recent approach is [3] that studied approximation to density function of the product using three methods: numerical integration, Monte Carlo simulation and analytical approximation to the result using the normal distribution. They showed as the inverse variation coefficient μ/σ increases, the distribution of the product of two independent normal variables tends towards a normal distribution. Our study is focused in Ware and Lad approaches. The objective was studying which factors have more influence in the presence of normality for the product of two independent normal variables. We have considered two factors: the inverse of the variation coefficient value μ/σ and the combined ratio (product of the two means divided by standard deviation): μ₁μ₂/σ for two normal variables with the same variance. Our results showed that for low values of the inverse of the variation coefficient (less than 1) normal distribution is not a good approximation for the product. Another one, influence of the combined ratio value is less than influence of the inverse of coefficients of variation value.
LA - eng
KW - product of normally distributed variables; inverse coefficient of variation; numerical integration; Monte Carlo simulation; combined ratio; normal distribution
UR - http://eudml.org/doc/271027
ER -

## References

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1. [1] Cecil C. Craig, On the frequency function of xy, Annals of Mathematical Society 7 (1936) 1-15. doi: 10.1214/aoms/1177732541. Zbl0013.35904
2. [2] L.A. Aroian, The probability function of a product of two normal distributed variables, Annals of Mathematical Statistics 18 (1947) 256-271. doi: 10.1214/aoms/1177730442. Zbl0041.45004
3. [3] R. Ware and F. Lad, Approximating the Distribution for Sums of Product of Normal Variables. Research-Paper 2003-15. Department of Mathematics and Statistics (University of Canterbury - New Zealand, 2003).
4. [4] L.A. Aroian, V.S. Taneja and L.W. Cornwell, Mathematical forms of the distribution of the product of two normal variables, Communication in Statistics - Theory and Method 7 (1978) 164-172. Zbl0385.62016
5. [5] J. Whisart and M.S.Bartlett, The distribution of second order moment statistics in a normal system, Proceedings of the Cambridge Philosophical Society XXVIII (1932) 455-459. doi: 10.1017/S0305004100010690. Zbl0005.40601
6. [6] L.A. Aroian, V.S. Taneja and L.W. Cornwell, Mathematical forms of the distribution of the product of two normal variables, Communications in Statistics. Theoretical Methods A7 (2) (1978) 165-172. doi: 10.1080/03610927808827610. Zbl0385.62016
7. [7] S.C. Chapra and R.P. Canale, Numerical Methods for Engineers (McGraw-Hill: New York, 2010).

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