Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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CHARACTERIZATIONS OF THE GAMMA DISTRIBUTION BY INDEPENDENCE PROPERTY OF RANDOM VARIABLES

  • Received : 2013.01.05
  • Accepted : 2013.04.04
  • Published : 2014.05.15
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Abstract

Let {$X_i$, $1{\leq}i{\leq}n$} be a sequence of i.i.d. sequence of positive random variables with common absolutely continuous cumulative distribution function F(x) and probability density function f(x) and $E(X^2)$ < ${\infty}$. The random variables X + Y and $\frac{(X-Y)^2}{(X+Y)^2}$ are independent if and only if X and Y have gamma distributions. In addition, the random variables $S_n$ and $\frac{\sum_{i=1}^{m}(X_i)^2}{(S_n)^2}$ with $S_n=\sum_{i=1}^{n}X_i$ are independent for $1{\leq}m$ < n if and only if $X_i$ has gamma distribution for $i=1,{\cdots},n$.

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References

  1. P. Findeisen, A simple proof of a classical theorem which characterizes the gamma distribution, Ann. Stat. Math. 6 (1978), no. 5, 1165-1167. https://doi.org/10.1214/aos/1176344319
  2. T. Y. Hwang and C. Y. Hu, On a characterizations of the gamma distribution: The independence of the sample mean and the sample coefficient of variation, Ann. Inst. Stat. Math. 51 (1999), no. 4, 749-753. https://doi.org/10.1023/A:1004091415740
  3. M. Y. Lee and E. H. Lim, A Characterization of gamma distribution by independent property, J. Chungcheong Math. soc. 22 (2009), no. 1, 1-5.
  4. E. Lukacs, A Characterization of gamma distribution, Ann. Math. Stat. 17 (1955), 319-324.
  5. E. Lukacs and R. G. Laha, Applications of characteristic function, Charles Griffin. London, 1963.

Cited by

  1. CHARACTERIZATIONS OF GAMMA DISTRIBUTION VIA SUB-INDEPENDENT RANDOM VARIABLES vol.28, pp.2, 2015, https://doi.org/10.14403/jcms.2015.28.2.187