scholarly journals Exponential and Hypoexponential Distributions: Some Characterizations

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2207
Author(s):  
George P. Yanev
Keyword(s):  
Linear Combination ◽  
Exponential Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Unknown Distribution ◽  
Converse Result ◽  
Hypoexponential Distribution ◽  
Exponential Random Variables

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some n≥2, X1,X2,…,Xn are independent copies of a random variable X with unknown distribution F and a specific linear combination of Xj’s has hypoexponential distribution, then F is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villaseñor (2013) for a particular convolution of two random variables.

scholarly journals The general theory of canonical correlation and its relation to functional analysis

1961 ◽  
Vol 2 (2) ◽  
pp. 229-242 ◽  
Author(s):  
E. J. Hannan
Keyword(s):  
Linear Combination ◽  
Canonical Correlation ◽  
Random Variables ◽  
Random Variable ◽  
Finite Variance ◽  
Finite Sample ◽  
Standard Description ◽  
Joint Distributions ◽  
Gaussian Case ◽  
The Relationship

The classical theory of canonical correlation is concerned with a standard description of the relationship between any linear combination of ρ random variablesxs, and any linear combination ofqrandom variablesytinsofar as this relation can be described in terms of correlation. Lancaster [1] has extended this theory, forp=q= 1, to include a description of the correlation of any function of a random variablexand any function of a random variabley(both functions having finite variance) for a class of joint distributions ofxandywhich is very general. It is the purpose of this paper to derive Lancaster's results from general theorems concerning the spectral decomposition of operators on a Hilbert space. These theorems lend themselves easily to the generalisation of the theory to situations wherepandqare not finite. In the case of Gaussian, stationary, processes this generalisation is equivalent to the classical spectral theory and corresponds to a canonical reduction of a (finite) sample of data which is basic. The theory also then extends to any number of processes. In the Gaussian case, also, the present discussion-is connected with the results of Gelfand and Yaglom [2] relating to the amount of information in one random process about another.

EQUIVALENT CHARACTERIZATIONS ON ORDERINGS OF ORDER STATISTICS AND SAMPLE RANGES

2010 ◽  
Vol 24 (2) ◽  
pp. 245-262 ◽  
Author(s):  
Tiantian Mao ◽  
Taizhong Hu
Keyword(s):  
Order Statistics ◽  
Exponential Distribution ◽  
Geometric Distribution ◽  
Random Variables ◽  
The Other ◽  
Stochastic Orders ◽  
Exponential Random Variables ◽  
Discrete Counterpart ◽  
Similarities And Differences ◽  
Geometric Variables

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

Probability Density Function of the Product and Quotient of Two Correlated Exponential Random Variables

1986 ◽  
Vol 29 (4) ◽  
pp. 413-418 ◽  
Author(s):  
Henrick J. Malik ◽  
Roger Trudel
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Exponential Distribution ◽  
Random Variables ◽  
Type I ◽  
Type Ii ◽  
Exponential Distributions ◽  
Bivariate Exponential Distribution ◽  
Bivariate Exponential ◽  
Exponential Random Variables

AbstractThis article deals with the distributions of the product and the quotient of two correlated exponential random variables. We consider here three types of bivariate exponential distributions: Marshall-Olkin's bivariate exponential distribution, Gumbel's Type I bivariate exponential distribution, and Gumbel's Type II bivariate exponential distribution.

A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1)

1977 ◽  
Vol 9 (1) ◽  
pp. 87-104 ◽  
Author(s):  
P. A. Jacobs ◽  
P. A. W. Lewis
Keyword(s):  
Linear Combination ◽  
Point Process ◽  
Moving Average ◽  
Random Variables ◽  
Stationary Sequence ◽  
Autoregressive Moving Average ◽  
Marginal Distributions ◽  
Joint Distributions ◽  
Exponential Random Variables ◽  
Exponential Sequence

A stationary sequence of random variables with exponential marginal distributions and the correlation structure of an ARMA (1, 1) process is defined. The process is formed as a random linear combination of i.i.d. exponential random variables and is very simple to generate on a computer. Moments and joint distributions for the sequence are obtained, as well as limiting properties of sums of the random variables and of the point process whose intervals have the EARMA (1, 1) structure.

Computation of Density for A Linear Combination of –Generalized Normal Random Variables

1975 ◽  
Vol 24 (1-4) ◽  
pp. 101-116
Author(s):  
Ru-Ying Lee ◽  
I. R. Goodman
Keyword(s):  
Linear Combination ◽  
Power Series Expansion ◽  
Random Variables ◽  
Random Variable ◽  
Computational Procedure ◽  
Normal Random Variable ◽  
Characteristic Functions ◽  
Accuracy Control ◽  
Truncated Form ◽  
Fourier Inversion

A computational procedure is presented for the approximation of the density of a linear combination of univariate -generalized normal random variables. (The -generalized normal random variable generalizes the ordinary normal one by replacing the power two in the exponent of the density by an arbitrary positive number.) The procedure applies a truncated form of the Fourier Inversion Theorem to the power series expansion of the characteristic function of a -generalized normal random variable. Because of the unimodal nature of -generalized normal characteristic functions for ⩽ 2 and the oscillatory nature for > 2, much of the computational procedure divides into two corresponding parts. Complete error analysis and accuracy control in all computations are also presented.

ON SKEWNESS AND DISPERSION AMONG CONVOLUTIONS OF INDEPENDENT GAMMA RANDOM VARIABLES

2010 ◽  
Vol 25 (1) ◽  
pp. 55-69 ◽  
Author(s):  
Leila Amiri ◽  
Baha-Eldin Khaledi ◽  
Francisco J. Samaniego
Keyword(s):  
Residual Life ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Mean Residual Life ◽  
Exponential Random Variables ◽  
Right Spread Order ◽  
New Better Than Used ◽  
Common Shape ◽  
Better Than

Let {x(1)≤···≤x(n)} denote the increasing arrangement of the components of a vector x=(x1, …, xn). A vector x∈Rn majorizes another vector y (written $\bf{x} \mathop{\succeq}\limits^{m} \bf{y}$) if $\sum_{i=1}^{j} x_{(i)} \le \sum_{i=1}^{j}y_{(i)}$ for j = 1, …, n−1 and $\sum_{i=1}^{n}x_{(i)} = \sum_{i=1}^{n}y_{(i)}$. A vector x∈R+n majorizes reciprocally another vector y∈R+n (written $\bf{x} \mathop{\succeq}\limits^{rm} \bf{y}$) if $\sum_{i=1}^{j}(1/x_{(i)}) \ge \sum_{i=1}^{j}(1/y_{(i)})$ for j = 1, …, n. Let $X_{\lambda_{i},\alpha},\,i=1,\ldots,n$, be n independent random variables such that $X_{\lambda_{i},\alpha}$ is a gamma random variable with shape parameter α≥1 and scale parameter λi, i = 1, …, n. We show that if $\lambda \mathop{\succeq}\limits^{rm} \lambda^{\ast}$, then $\sum_{i=1}^{n} X_{\lambda_{i},\alpha}$ is greater than $\sum_{i=1}^{n} X_{\lambda^{\ast}_{i},\alpha}$ according to right spread order as well as mean residual life order. We also prove that if $(1/ \lambda_{1}, \ldots ,1/ \lambda_{n}) \mathop{\succeq}\limits^{m} \succeq (1/ \lambda_{1}^{\ast}, \ldots , 1/ \lambda_{n}^{\ast})$, then $\sum_{i=1}^{n} X_{\lambda_{i}, \alpha}$ is greater than $\sum_{i=1}^{n} X_{\lambda^{\ast}_{i},\alpha}$ according to new better than used in expectation order as well as Lorenze order. These results mainly generalize the recent results of Kochar and Xu [7] and Zhao and Balakrishnan [14] from convolutions of independent exponential random variables to convolutions of independent gamma random variables with common shape parameters greater than or equal to 1.

scholarly journals Some Characterizations of Exponential Distribution

2017 ◽  
Vol 6 (5) ◽  
pp. 132
Author(s):  
M. Ahsanullah ◽  
M. Z. Anis
Keyword(s):  
Distribution Function ◽  
Linear Combination ◽  
Lebesgue Measure ◽  
Exponential Distribution ◽  
Random Variables ◽  
Absolutely Continuous ◽  
Measure Distribution ◽  
Independent And Identically Distributed

There are some characterizations of the exponential distribution based on the relation of the maximum of two observations expressed as linear combination of the two observations. In this paper some generalizations of this known characterization of the exponential distribution using the relations between the maximum and minimum of  independent and identically distributed random variables having absolutely continuous (with respect to Lebesgue measure) distribution function will be presented.

scholarly journals DETERMINATION OF VARIANCE OF TIME TO RECRUITMENT WITH INTER-DECISION TIME AS GEOMETRIC PROCESS WHEN THE THRESHOLD HAS THREE COMPONENTS

YMER Digital ◽  
2021 ◽  
Vol 20 (11) ◽  
pp. 222-229
Author(s):  
A DEVI ◽  
◽  
B SATHISH KUMAR ◽  
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Decision Time ◽  
Breakdown Threshold ◽  
Geometric Process ◽  
Exponential Random Variables ◽  
Manpower System ◽  
Three Components ◽  
Independent And Identically Distributed

In this paper, the problem of time to recruitment is analyzed for a single grade manpower system using an univariate CUM policy of recruitment. Assuming policy decisions and exits occur at different epochs, wastage of manpower due to exits form a sequence of independent and identically distributed exponential random variables, the inter-decision times form a geometric process and inter-exist time form an independent and identically distributed random variable. The breakdown threshold for the cumulative wastage of manpower in the system has three components which are independent exponential random variables. Employing a different probabilistic analysis, analytical results in closed form for system characteristics are derived

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