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Notes on Exponential Distribution

المتغير العشوائي X  له توزيع أسي اذا كان له دالة احتمالية الكثافة بالشكل:

عندما  ، هذه هي الحالة الخاصة لتوزيع كاما.

غالباً جداً ولسبب معقول تأخذ . الحالة الخاصة لـ (1) التي نحصل عليها تسمى بالتوزيع الاسي لمعلمة واحدة.

اذا كانت  ، ، التوزيع في هذه الحالة يسمى التوزيع الاسي القياسي

اما بالنسبة لعائلة التوزيع الاسي.

افرض ان                                             

هي دالة لمعلمة واحدة لعائلة دوال احتمالية الكثافة الاسية بالنسبة الى قياس Lebesgue على مجموعات Borel لـ HCR مع معلمة الفضاء  التي تكتب بالشكل:

 

افرض ان  وان b هي قابلة للاشتقاق

الحالة  مع  هي اعتيادياً تعود الى الشكل القياسي ولاجل دالة احتمالية قياس لتوزيع اسي لشكل قياس c (.) ، d(.) وحيدة وتؤدي الى تحويلات

 انياً لاجل  وهكذا فاننا نفرض ان c(o) = o بدون فقدان التعميم.

 

 

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Publication Date
Tue Feb 13 2024
Journal Name
Iraqi Journal Of Science
Suggested method of Estimation for the Two Parameters of Weibull Distribution Using Simulation Technique

In this paper, suggested method as well as the conventional methods (probability
plot-(p.p.) for estimations of the two-parameters (shape and scale) of the Weibull
distribution had proposed and the estimators had been implemented for different
sample sizes small, medium, and large of size 20, 50, and 100 respectively by
simulation technique. The comparisons were carried out between different methods
and sample sizes. It was observed from the results that suggested method which
were performed for the first time (as far as we know), by using MSE indicator, the
comparisons between the studied and suggested methods can be summarized
through extremely asymptotic for indicator (MSE) results by generating random
error

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Publication Date
Thu Oct 31 2013
Journal Name
Al-khwarizmi Engineering Journal
An Investigation Study of Thinning Distribution in Single Point Incremental Forming Using FEM Analysis

Single Point Incremental Forming (SPIF) is a forming technique of sheet material based on layered manufacturing principles. The sheet part is locally deformed through horizontal slices. The moving locus of forming tool (called as toolpath) in these slices constructed to the finished part was performed by the CNC technology. The toolpath was created directly from CAD model of final product. The forming tool is a Ball-end forming tool, which was moved along the toolpath while the edges of sheet material were clamped rigidly on fixture.

This paper presented an investigation study of thinning distribution of a conical shapes carried out by incremental forming and the validation of finite element method to evaluate the limits of the p

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Publication Date
Sun Jun 11 2017
Journal Name
Ibn Al-haitham Journal For Pure And Applied Sciences
An Efficient Single Stage Shrinkage Estimator for the Scale parameter of Inverted Gamma Distribution

 The present  paper agrees  with estimation of scale parameter θ of the Inverted Gamma (IG) Distribution when the shape parameter α is known (α=1), bypreliminarytestsinglestage shrinkage estimators using  suitable  shrinkage weight factor and region.  The expressions for the Bias, Mean Squared Error [MSE] for the proposed estimators are derived. Comparisons between the considered estimator with the usual estimator (MLE) and with the existing estimator  are performed .The results are presented in attached tables.

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Publication Date
Thu Dec 01 2011
Journal Name
Iraqi Journal Of Physics
Design of Achromatic Combined Quadrupole Lens Using the Modified Bell-Shaped Field Distribution Model

The optimization calculations are made to find the optimum properties of combined quadrupole lens consist of electrostatic and magnetic lenses to produce achromatic lens. The modified bell-shaped model is used and the calculation is made by solving the equation of motion and finding the transfer matrices in convergence and divergence planes, these matrices are used to find the properties of lens as the magnification and aberrations coefficients. To find the optimum values of chromatic and spherical aberrations coefficients, the effect of both the excitation parameter of the lens (n) and the effective length of the lens into account as effective parameters in the optimization processing

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Publication Date
Tue Dec 01 2009
Journal Name
Journal Of Economics And Administrative Sciences
Estimation of the parameter of the Pareto distribution manual Using the general mediator estimator

Estimation of the tail index parameter of a one - parameter Pareto model has wide important by the researchers because it has awide application in the econometrics science and reliability theorem.

Here we introduce anew estimator of "generalized median" type and compare it with the methods of Moments and Maximum likelihood by using the criteria, mean square error.

 The estimator of generalized median type performing best over all.

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Publication Date
Mon Jun 19 2023
Journal Name
Journal Of Engineering
An Analysis of Stress Distribution in a Spline Shaft Subjected to Cycilc Impulsive Load

In this paper the effect of engagement length, number of teeth, amount of applied load, wave propagation time, number of cycles, and initial crack length on the principal stress distribution, velocity of crack propagation, and cyclic crack growth rate in a spline coupling subjected to cyclic torsional impact have been investigated analytically and experimentally. It was found that the stresses induced due to cyclic impact loading are higher than the stresses induced due to impact loading with high percentage depends on the number of cycles and total loading time. Also increasing the engagement length and the number of teeth reduces the principal stresses (40%) and
(25%) respectively for increasing the engagement length from (0.15 to 0

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Publication Date
Tue Dec 31 2019
Journal Name
Journal Of Economics And Administrative Sciences
Comparing Different Estimators for the shape Parameter and the Reliability function of Kumaraswamy Distribution

In this paper, we used maximum likelihood method and the Bayesian method to estimate the shape parameter (θ), and reliability function (R(t)) of the Kumaraswamy distribution with two parameters l , θ (under assuming the exponential distribution, Chi-squared distribution and Erlang-2 type distribution as prior distributions), in addition to that we used method of moments for estimating the parameters of the prior distributions. Bayes

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Publication Date
Fri Dec 30 2022
Journal Name
Journal Of Mathematics
Estimation of Parameters of Finite Mixture of Rayleigh Distribution by the Expectation-Maximization Algorithm

In the lifetime process in some systems, most data cannot belong to one single population. In fact, it can represent several subpopulations. In such a case, the known distribution cannot be used to model data. Instead, a mixture of distribution is used to modulate the data and classify them into several subgroups. The mixture of Rayleigh distribution is best to be used with the lifetime process. This paper aims to infer model parameters by the expectation-maximization (EM) algorithm through the maximum likelihood function. The technique is applied to simulated data by following several scenarios. The accuracy of estimation has been examined by the average mean square error (AMSE) and the average classification success rate (ACSR). T

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Publication Date
Sun Dec 29 2019
Journal Name
Iraqi Journal Of Science
Distribution of Polycyclic Aromatic Hydrocarbons in Water and Sediments in the Euphrates River, Iraq

The monitoring of lotic ecosystems is important for environmental health after war crisis. This study aimed to detect the fate of the sixteen polycyclic aromatic hydrocarbon compounds (PAHs) as priority pollutants in the water and sediment of Al-Hussainya River in Karbala Province. The results indicated that the concentrations in water samples ranged from 0.24–58.72 ηg.L-1 for each of the benzo(a) pyrene and benzo(g, h, i)perylene. The concentrations in sediment samples have been observed to be in a range of 0.36–119.06 μg.g-1 for naphthalene and benzo(g,h,i)perylene. Benzo(g,h,i)perylene recorded the highest concentrations in each of water and sediment samples as compared with the other compound

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Publication Date
Thu Mar 30 2023
Journal Name
Iraqi Journal Of Science
Acute HAV Infected Patient's Distribution According to Age and Sex in Wasit Province/Iraq.

Hepatitis A virus infection occurs globally and is causing a public health
concern, primarily in developing countries due to its persistent circulation in the
environment. A total of 120 individuals were included in this study, anti-HAV IgM
antibodies were detected in serum of 100 individuals of them (patients group) who
attended the Wasit central public health laboratory from different cities in Wasit
Province, Iraq, during the period from November 2013 to March 2014.While
another 20 individuals were considered as control group which anti – HAV IgM
antibody was negative in their serum.
Acute infection is confirmed by detection of IgM anti–hepatitis A virus (HAV),
which appears early in the course of infect

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