The Distribution of a Linear Combination of Two Correlated Chi-Square Variables

Distribución de una combinación lineal de dos variables chi-cuadrado correlacionadas

The distribution of the linear combination of two chi-square variables is known if the variables are independent. In this paper, we derive the distribution of positive linear combination of two chi-square variables when they are correlated through a bivariate chi-square distribution. Some properties of the distribution, namely, the characteristic function, cumulative distribution function, raw moments, mean centered moments, coefficients of skewness and kurtosis are derived. Results match with the independent case when the variables are uncorrelated. The graph of the density function is presented. 

La distribución de una combinación lineal de dos variables chi cuadrado es conocida si las variables son independientes. En este artículo, se deriva la distribución de una combinación lineal positiva de dos variables chi cuadrado cuando estas están correlacionadas a través de una distribución chi cuadrado bivariada. Algunas propiedades de esta distribución como la función característica, la función de distribución acumulada, sus momentos, momentos centrados alrededor de la media, los coeficientes de sesgo y curtosis son derivados. Los resultados coinciden con el caso independiente cuando las variables son no correlacionadas. La gráfica de la función de densidad es también presentada.

The Distribution of a Linear Combination ofTwo Correlated Chi-Square Variables Distribución de una combinación lineal de dos variables chi-cuadrado correlacionadas ANWAR H. JOARDER¹, M. HAFIDZ OMAR², ARJUN K. GUPTA³

¹King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran, Saudi Arabia. Professor. Email: anwarj@kfupm.edu.sa
²King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran, Saudi Arabia. Associate professor. Email: omarmh@kfupm.edu.sa
³Bowling Green State University, Department of Mathematics and Statistics, Ohio, United States of America. Professor. Email: gupta@bgsu.edu

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Abstract

Key words: Bivariate Chi-square Distribution, CorrelatedChi-square Variables, Linear Combination, Characteristic Function, Cumulative Distribution, Moments.

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Resumen

Palabras clave: combinación lineal, distribución acumulada, distribución chi cuadrado bivariada, función característica, momentos, variables chi cuadrado correlacionadas.

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Texto completo disponible en PDF

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References

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[Recibido en febrero de 2013. Aceptado en julio de 2013]

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv36n2a02,
    AUTHOR  = {Joarder, Anwar H. and Omar, M. Hafidz and Gupta, Arjun K.},
    TITLE   = {{The Distribution of a Linear Combination ofTwo Correlated Chi-Square Variables}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2013},
    volume  = {36},
    number  = {2},
    pages   = {209-219}
}

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