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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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참고문헌 (5)
참고문헌 신청
P. Findeisen / 1978 / A simple proof of a classical theorem which characterizes the gamma distribution / Ann. Stat. Math 6(5):1165~1167
T. Y. Hwang / 1999 / On a characterizations of the gamma distribution: The independence of the sample mean and the sample coefficient of variation / Ann. Inst. Stat. Math 51(4):749~753
Min-Young Lee / 2009 / A CHARACTERIZATION OF GAMMA DISTRIBUTION BY INDEPENDENT PROPERTY / 충청수학회지 22(1):1~5
E. Lukacs / 1955 / A Characterization of gamma distribution / Ann. Math. Stat 17:319~324
E. Lukacs / 1963 / Applications of characteristic function / Charles Griffin
T. Y. Hwang / 1999 / On a characterizations of the gamma distribution: The independence of the sample mean and the sample coefficient of variation / Ann. Inst. Stat. Math 51(4):749~753
Min-Young Lee / 2009 / A CHARACTERIZATION OF GAMMA DISTRIBUTION BY INDEPENDENT PROPERTY / 충청수학회지 22(1):1~5
E. Lukacs / 1955 / A Characterization of gamma distribution / Ann. Math. Stat 17:319~324
E. Lukacs / 1963 / Applications of characteristic function / Charles Griffin
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