• Journal of the Korean Data and Information Science Society
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• v.10 no.1
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• pp.243-250
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• 1999

We consider $Y=X{\lambda}Z,\;{\lambda}>0$, where X and Z are independent random variables, and Y is the length biased distribution or the equilibrium distribution of X. The purpose of this paper is to consider the distribution of X or Y when the distribution of Z is given and the distribution of Z when the distribution of X or Y is given, In particular, we obtain that the necessary and sufficient conditions for X to be $X^{2}({\upsilon})\;is\;Z{\sim}X^{2}(2)\;and\;for\;Z\;to\;be\;X^{2}(1)\;is\;X{\sim}IG({\mu},\;{\mu}^{2}/{\lambda})$, where $IG({\mu},\;{\mu}^{2}/{\lambda})$ is two-parameter inverse Gaussian distribution. Also we show that X is smaller than Y in the reverse Laplace transform ratio order if and only if $X_{e}$ is smaller than $Y_{e}$ in the Laplace transform ratio order. Finally, we can get the results that if X is smaller than Y in the Laplace transform ratio order, then $Y_{L}$ is smaller than $X_{L}$ in the Laplace transform order, and that if X is smaller than Y in the reverse Laplace transform ratio order, then $_{\mu}X_{L}$ is smaller than $_{\nu}Y_{L}$ in the Laplace transform order.

• Journal of the Korean Statistical Society
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• v.17 no.1
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• pp.46-55
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• 1988

Two-piece double exponential distribution (TPDE) with one piece $(X \leq 0)$ having the scale parameter $\theta_1$ while the other piece (X>0) having $\theta_2$ is considered here. Distribution of the sum of n-independent variables from such a distribution is obtained. Special cases of this distribution are also treated. Next, distribution of the ratio of two independent (TPDE) variables is derived. As an extension, distribution of $x_1/x_2x_3$ is expressed terms of hypergeometric functions. A small table gives the power of the test regarding double exponential against (TPDE).

• Journal of the Chungcheong Mathematical Society
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• v.22 no.1
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• pp.1-5
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• 2009

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.

• Journal of applied mathematics & informatics
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• v.35 no.3_4
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• pp.261-265
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• 2017

Let X and Y be independent identically distributed nondegenerate random variables with common absolutely continuous probability distribution function F(x) and the corresponding probability density function f(x) and $E(X^2)$<${\infty}$. Put Z = max(X, Y) and W = min(X, Y). In this paper, it is proved that Z - W and Z + W or$(X-Y)^2$ and X + Y are independent if and only if X and Y have normal distribution.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.2
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• pp.157-163
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• 2014

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 $i=1,{\cdots},n$.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.2
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• pp.245-250
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• 2010

We present characterizations of the Weibull distribution by the independent property of record values that F(x) has a Weibull distribution if and only if $\frac{X_{U(m)}}{X_{U(n)}}$ and $X_{U(n)}$ or $\frac{X_{U(n)}}{X_{U(n)}{\pm}X_{U(m)}}$ and $X_{U(n)}$ are independent for $1{\leq}m.

• Communications of the Korean Mathematical Society
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• v.18 no.1
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• pp.127-131
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• 2003

Let X$_1$, X$_2$,... be a sequence of independent and identically distributed random variables with continuous cumulative distribution function F(x). X$_j$ is an upper record value of this sequence if X$_j$ > max {X$_1$,X$_2$,...,X$_{j-1}$}. We define u(n)=min{j$\mid$j> u(n-1), X$_j$ > X$_{u(n-1)}$, n $\geq$ 2} with u(1)=1. Then F(x) = 1-x$^{\theta}$, x > 1, ${\theta}$ < -1 if and only if (${\theta}$+1)E[X$_{u(n+1)}$$\mid$X$_{u(m)}$=y] = ${\theta}E[X_{u(n)}$\mid$X_{u(m)}=y], (\theta+1)^2E[X_{u(n+2)}$\mid$X_{u(m)}=y] = \theta^2E[X_{u(n)}$\mid$X_{u(m)}=y], or (\theta+1)^3E[X_{u(n+3)}$\mid$X_{u(m)}=y] = \theta^3E[X_{u(n)}$\mid$X_{u(m)}=y], n $\geq$ M+1$.

• Honam Mathematical Journal
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• v.26 no.4
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• pp.463-469
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• 2004

In this paper we establish some recurrence relations satisfied by quotient moments of upper record values from the exponential distribution. Let $\{X_n,\;n{\geq}1\}$ be a sequence of independent and identically distributed random variables with a common continuous distribution function F(x) and probability density function(pdf) f(x). Let $Y_n=max\{X_1,\;X_2,\;{\cdots},\;X_n\}$ for $n{\geq}1$. We say $X_j$ is an upper record value of $\{X_n,\;n{\geq}1\}$, if $Y_j>Y_{j-1}$, j > 1. The indices at which the upper record values occur are given by the record times {u(n)}, $n{\geq}1$, where u(n)=min\{j{\mid}j>u(n-1),\;X_j>X_{u(n-1)},\;n{\geq}2\} and u(1) = 1. Suppose $X{\in}Exp(1)$. Then $\Large{E\;\left.{\frac{X^r_{u(m)}}{X^{s+1}_{u(n)}}}\right)=\frac{1}{s}E\;\left.{\frac{X^r_{u(m)}}{X^s_{u(n-1)}}}\right)-\frac{1}{s}E\;\left.{\frac{X^r_{u(m)}}{X^s_{u(n)}}}\right)}$ and $\Large{E\;\left.{\frac{X^{r+1}_{u(m)}}{X^s_{u(n)}}}\right)=\frac{1}{(r+2)}E\;\left.{\frac{X^{r+2}_{u(m)}}{X^s_{u(n-1)}}}\right)-\frac{1}{(r+2)}E\;\left.{\frac{X^{r+2}_{u(m-1)}}{X^s_{u(n-1)}}}\right)}$.

• Communications of the Korean Mathematical Society
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• v.16 no.2
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• pp.287-290
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• 2001

Let X$_1$, X$_2$, … be a sequence of independent and identically distributed random variables with continuous cumulative distribution function F(x). X(sub)j is an upper record value of this sequence if X(sub)j > max {X$_1$, X$_2$, …, X(sub)j-1}. We define u(n) = min {j│j > u(n-1), X(sub)j > X(sub)u(n-1), n $\geq$ 2} with u(1) = 1. Then F(x) = 1 - e(sup)-x/c, x > 0 if and only if E[X(sub)n(n+1) - X(sub)u(n)│X(sub)u(m) = y] = c or E[X(sub)u(n+2) - X(sub)u(n)│X(sub)u(m) = y] = 2c, n $\geq$ m+1.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.4
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• pp.411-418
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• 2007

Let $X_1$, ${\cdots}$, $X_n$ be nondegenerate and positive independent identically distributed(i.i.d.) random variables with common absolutely continuous distribution function F(x) and $E(X^2)$ < ${\infty}$. The random variables $X_1+{\cdots}+X_n$ and $\frac{X_1+{\cdots}+X_m}{X_1+{\cdots}+X_n}$are independent for 1 $1{\leq}$ m < n if and only if $X_1$, ${\cdots}$, $X_n$ have gamma distribution.