충청수학회지 (Journal of the Chungcheong Mathematical Society)

  • 제22권1호
  • /
  • Pages.1-5
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  • 2009
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  • 1226-3524(pISSN)
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  • 2383-6245(eISSN)
충청수학회 (The Chungcheong Mathematical Society)
권호 표지

A CHARACTERIZATION OF GAMMA DISTRIBUTION BY INDEPENDENT PROPERTY

  • 투고 : 2008.09.25
  • 심사 : 2009.02.13
  • 발행 : 2009.03.31
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초록

Let {$X_n,\;n{\geq}1}$ be a sequence of independent identically distributed(i.i.d.) sequence of positive random variables with common absolutely continuous distribution function(cdf) F(x) and probability density function(pdf) f(x) and $E(X^2)<{\infty}$. The random variables $\frac{X_i{\cdot}X_j}{(\Sigma^n_{k=1}X_k)^{2}}$ and $\Sigma^n_{k=1}X_k$ are independent for $1{\leq}i if and only if {$X_n,\;n{\geq}1}$ have gamma distribution.

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참고문헌

  1. P. Findeisen, A simple proof of a classical theorem which characterizes the gamma distribution, Ann. Stat. 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, Characterization of gamma distribution, J. Chungcheong Math. soc. 20 (2007), 411-418.
  4. E. Lukacs, A Characterization of the gamma distribution, Ann. Math. Stat. 17 (1955), 319-324.