### The Distribution Properties of Two-Parameter Exponential Distribution Order Statistics

#### Article Information

**Roland Forson ^{1*}, Cai Guanghui^{1}, Samuel Ofori^{2}, Oo Than Nweit^{2}, Daniel Ofori Kusi^{3}**

^{1}Department of Statistics and Mathematics, Zhejiang Gongshan University, China

^{2}Department of Mathematics and Physics, Zhejiang Normal University, China

^{3}Department of Mathematics, University of Free State, South Africa

***Corresponding Author: **Roland Forson, Department of Statistics and Mathematics, Zhejiang Gongshan University, China

**Received: **25 April 2019;** Accepted: **04May 2019;** Published: **10 May 2019

**Citation:** Roland Forson, Cai Guanghui, Samuel Ofori, Oo Than Nweit, Daniel Ofori Kusi. The Distribution Properties of Two-Parameter Exponential Distribution Order Statistics. Journal of Analytical Techniques and Research 1 (2019): 003-009.

#### Abstract

This paper proposes the distribution function and density function of double parameter exponential distribution and discusses some important distribution properties of order statistics. We prove that random variables following the double parameter exponential type distribution X1, X2,..., Xn are not mutually independent and do not follow the same distribution, but that the Xi, Xj meet the dependency of TP2 to establish RTI ( Xi | Xj ), LTD (Xi | Xj ) and RSCI.

#### Keywords

Order statistics, Double parameter exponential distribution, TP2, RTI, LTD, RSCI

#### Article Details

#### 1. Introduction

In this paper, some important properties of order statistics of two-parameter exponential distribution are discussed when the distribution and density functions of a two-parameter distribution is given. We also proved that the random variables *X*_{1},* X*_{2},...,* X _{n}*, obeying the two-parameter exponential distribution are not independent of each other, and do not obey the same distribution. Order statistics is a kind of statistics distribution commonly used in statistical theory and application of which there are many research [1-6]. The two parameter exponential distribution is also a very useful component in reliability engineering. This study considers the nature of order statistics. Its density function and distribution functions are respectively [7];

#### 2. Prerequisite Knowledge

**2.1 Lemma 1**

Let all X follow a continuous distribution function *F (x)* and its density function of* F (x), {a < x < b}, X _{1}, X_{2},...X_{n} *is a simple random sample with acapacity of

*N*from

*X*

_{2}- The joint probability density function of
*(X*is_{1}, X_{2},..., X_{n}) if a ≤ X_{1 }< X_{ 2}< ... < X_{n }≤ b

Otherwise,

*g*_{1,2,...,n} (* X*_{1}*, X *_{2}*, *...,* X _{n} *) 0

- The joint probability density function of order statistic (
*Xi, Xj*)(1*≤**I**≤**j**≤**n*) is a*≤ x ≤ y ≤ b*and

Otherwise,

*g _{i}*

_{,}

*(*

_{j}*x*,

*y*) 0

- The probability density function of order statistics
*X (k)*is

In particular, when *k = 1*, there is

When *k = n, *there is

**2.2 Lemma 2**

Assume that *X *and *Y* are two random variables, the joint probability is *f (x)* if the inequality *x _{1} ≤ x_{2}, y_{1}≤ y_{2} *is satisfied [10].

**2.3 Lemma 3**

For a fixed *x*, *y*, if *P* (*X > x*, *Y > y *| *X > x*^{1}*, Y > y*^{1} ) is a monotonic increasing function for variables *x*^{1} and *y*^{1}, then variables* X, Y *satisfies RSCI [10] .

**2.4 Lemma 4**

For any *y*_{1}, if* P *(*Y ≤ y*_{1}| *X ≤ x*_{1} ) is monotone decreasing function of* x*_{1}*, *then* Y *is the left tail decreasing function of *X, *denoted by* Ltd *(* XY *) [11].

**2.5 Lemma 5**

For any *y*_{1}, if* P *(*Y > y*_{1} | *X > x*_{1} ) is an incremental function of* x*_{1}, then *Y *is the right tail growth of* X, *denoted by *RTI *(*XY*) [11]*.*

#### 3. Main Conclusion

**3.1 Theorem 1**

Let the total* X* follow a continuous distribution function of* F *(*x*) and its density function* F *(*x*) (*a < x < b*),* X*_{1},* X*_{2},...,* X _{n} *be a simple random sample with a capacity of

*N*from the population

*X*, and

*X*

_{1},

*X*

_{2},...,

*X*

_{n}- The order statistics of the joint probability density function of (
*X*_{1},*X*_{2},...,*X*) is as follows;_{n}*a ≤ X*_{1}*<X*_{2 }*<*...*<X*_{n}*≤ b*

If then

- The probability density function of
*X*is;_{k}

In particular, when* k *1, there is

When *k= n*,

Which is proved from lemma 1.

**3.2 Theorem 2**

If *X*_{1}, *X*_{2}**, **..., *X _{n}* is independent of each other and obeys a two-parameter exponential distribution, then

*X*

_{1},

*X*

_{2}, ...,

*X*is not independent of each other and does not obey the same distribution.

_{n}*Proof : *Let* n = *2 be used to represent the observations of* x*_{1} and* x*_{2} with* X*_{1} and* X*_{2} respectively. It can be seen from lemma 1 that the density function of (*X*_{1}, *X*_{2}) is

The density functions of *X*_{1} and *X *_{2} are respectively

Assuming that *X*_{1} and *X* _{2} are independent, then *f _{X}*

_{1,X}

_{2}(

*x*

_{1},

*x*

_{2}) =

*f*(

*x*

_{1}).

*f*(

*x*

_{2}), and 2

*f*(

*x*

_{1})

*f*(

*x*

_{2}) = 2[1-

*F*(

*x*

_{1})]

*f*(

*x*

_{1}).2

*F*(

*x*

_{2})

*f*(

*x*

_{2}) = 4 [1-

*F*(

*x*

_{1})]

*F*(

*x*

_{2})

*f*(

*x*

_{1})

*f*(

*x*

_{2}).

Therefore,

Then,

Which obviously does not hold. So

*f *_{X}_{1}_{,} _{X}_{2} (*x*_{1},* x*_{2} ) ≠ *f *(*x*_{1} ).* f *(*x*_{2})

Therefore, *X*_{1} and *X* _{2} are not independent of each other and do not obey the same distribution. Hence proved.

**3.3 Theorem 3**

If *X*_{1}*, X*_{2},...,* X _{n} *are independent of each other and obey the two-parameter exponential distribution, then

*X i*,

*Yj*(

*I < j*) is

*TP*2 dependent.

*Proof: *The observations of *xi *and* xj* are represented by* Xi*,* Xj. *For any *x _{i }*<

*x*the joint density function of Lemma 1, is

_{j},We only need

**3.4 Theorem 4**

Suppose X_{1}, X_{2}, ..., X_{n} is* n *independent and identically distributed random variables from the exponential distribution of two parameters, then *RTI (X _{j} | X_{i}), i< j*. It is proved that for any

*s, t*when

*s > t*, there is

When *it* is fixed and *so *increases. The integral of *P (≤ X _{j }≤ t, Xs)* decreases

*P*(

*s ≤ X*)

_{j}≤ t, X_{i}> s*P *(*X _{j} > t | Xi > s) LTD( Xi | X_{j}), i < j* according to the density functions becomes smaller and becomes larger.

**3.5 Theorem 5**

Suppose *X _{i} *and

*X*are two random variables independently obeying the same two-parameter exponent. Then

_{j}*X*and

_{1}*X*satisfy

_{2}*RCSI*(i.e for fixed

*X*and

*Y*and

*Y*, if

*P*(

*X x,Y y | X x ', Y y'*) is a monotonic increasing function of variables x' and

*y'*, it is proved that the joint density functions of

*X*and

_{1}*X*is

_{2}*f*(

_{1,2 }*x*). When

_{1}, x_{2}*x*is invariant,

_{1, }x_{2}*P*(

*X*>

_{1 }> x_{1}, X_{2}> x_{2}| X_{1}> x'_{1}, X_{2}*x'*),

_{2}*x'*is increasing function

_{1}, x'_{2}iii When *x _{1} < x'_{1}, x_{2} > x'_{2}* or

*x*are equally available,

_{1}> x'x, x_{2}< x'_{2}When* x _{1, }x_{2 }*are fixed, they are incremental functions of x'

_{1}, x'

_{2}. In conclusion,

*P*(

*X*) are incremental functions of

_{1}> x_{1}, X_{2 }> x_{2}| X_{1}> x'_{1}, X_{2 }> x'_{2}*x*So

_{1}', x'_{2}.*X*satisfies

_{1, }X_{2}*RSCI.*

#### 4. Conclusion

Some distribution properties of order statistics obeying two-parameter exponential distribution are discussed. It is proved that when X_{1}, X_{2},..., X_{n} are independent of each other and obey the exponential distribution of two-parameters, the order statistics X_{1}, X_{2}, ..., X_{n }is not independent of each other and does not obey the same distribution, but X_{i}, X_{j} satisfies *TP2 *dependence. For any *i < j*, there are *RTI* ( *X _{j} | X_{i }*),

*LTD*( X

_{i }| X

_{j}), and X

_{1, }X

_{2}that satisfy

*RSCI*.

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