scholarly journals SIFAT-SIFAT DAN KEJADIAN KHUSUS DISTRIBUSI GAMMA

2021 ◽  
Vol 15 (1) ◽  
pp. 047-058
Author(s):  
Royke Yohanes Warella ◽  
Henry Junus Wattimanela ◽  
Venn Yan Ishak Ilwaru
Keyword(s):  
Gamma Distribution ◽  
Shape Parameter ◽  
Random Variable ◽  
Moment Generating Function ◽  
Continuous Random Variable ◽  
Positive Real ◽  
Chi Square ◽  
Special Cases ◽  
Distribution Parameters ◽  
The Moment

The gamma distribution is one of special continuous random variable distribution with scale parameter  and shape parameter  where  is positive real numbers. On some conditions the gamma distribution astablishes other continuous distributions which are then called special cases of the gamma distribution. Therefore, this study was conducted to determine the properties of gamma distribution and the characteristics of the special cases of gamma distribution by analyzed the theories from literatures. The properties of gamma distribution include expectation value, variance, moment generating function, characteristic function, and estimation of gamma distribution parameters with the moment method to earn the special cases of the gamma distribution are Erlang, exponential, chi-square, and beta distributions.

On the linear exponential family

1968 ◽  
Vol 64 (2) ◽  
pp. 481-483 ◽  
Author(s):  
J. K. Wani
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Scalar Product ◽  
Exponential Family ◽  
Characterization Theorem ◽  
Random Variable ◽  
Moment Generating Function ◽  
The Moment ◽  
Image Position

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given bywhere a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

scholarly journals A Study of Properties and Applications of Gamma Distribution

2021 ◽  
Vol 4 (2) ◽  
pp. 52-65
Author(s):  
Eric U. ◽  
Oti M.O.O. ◽  
Francis C.E.
Keyword(s):  
Probability Distribution ◽  
Gamma Distribution ◽  
Generating Function ◽  
Real Life ◽  
Moment Generating Function ◽  
Reliability Theory ◽  
Burn Out ◽  
Electrical Devices ◽  
The Moment ◽  
Special Case

The gamma distribution is one of the continuous distributions; the distributions are very versatile and give useful presentations of many physical situations. They are perhaps the most applied statistical distribution in the area of reliability. In this paper, we present the study of properties and applications of gamma distribution to real life situations such as fitting the gamma distribution into data, burn-out time of electrical devices and reliability theory. The study employs the moment generating function approach and the special case of gamma distribution to show that the gamma distribution is a legitimate continuous probability distribution showing its characteristics.

Exact Distribution of the Quotient of Independent Generalized Gamma Variables

1967 ◽  
Vol 10 (3) ◽  
pp. 463-465 ◽  
Author(s):  
Henrick John Malik
Keyword(s):  
Gamma Distribution ◽  
Random Variable ◽  
Exact Distribution ◽  
Frequency Function ◽  
Generalized Gamma ◽  
Special Cases ◽  
Chi Squared ◽  
Image Position

Let X be a random variable whose frequency function is1.1Form (1.1) is Stacy′s [3] generalization of the gamma distribution. The familiar gamma, chi, chi-squared, exponential and Weibull variâtes are special cases, as are certain functions of normal variate - viz., its positive even powers, its modulus, and all positive powers of its modulus.

Probabilistic representations of matrix variate skew normal models

2015 ◽  
Vol 23 (1) ◽  
Author(s):  
Wei Ning
Keyword(s):  
Normal Distribution ◽  
Random Variable ◽  
Moment Generating Function ◽  
Normal Random Variable ◽  
Skew Normal Distribution ◽  
Probabilistic Representation ◽  
The Matrix ◽  
Distribution Family ◽  
The Moment ◽  
Skew Normal

AbstractAzzalini [Scand. J. Stat. 12 (1985), 171–178] first introduced the skew normal distribution family with a shape parameter λ, and then extended this family by adding an additional shape parameters ξ. Basic properties of these two families were studied. Henze [Scand. J. Stat. 13 (1986), 271–275] gave the probabilistic representations for these two families by interpreting it as the linear combination of a normal random variable with another normal random variable truncated at the origin and several properties were illustrated. Chen and Gupta [Statistics 39 (2005), no. 3, 247–253] extended the skew normal distribution family to the matrix variate and proposed the moment generating function and the quadratic form of the matrix variate skew normal models. Motivated by these results, we first study the probabilistic representation for the matrix variate skew normal models and several properties. Then we define the extended skew normal model of the matrix variate, and give the probabilistic representation for this family and its extension.

Inequalities with application in economic risk analysis

1972 ◽  
Vol 9 (2) ◽  
pp. 441-444 ◽  
Author(s):  
Robert A. Agnew
Keyword(s):  
Risk Analysis ◽  
Lower Bounds ◽  
Generating Function ◽  
Random Variable ◽  
Moment Generating Function ◽  
Upper Bounds ◽  
Economic Risk ◽  
The Moment

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

scholarly journals Constructing Matrix Exponential Distributions by Moments and Behavior around Zero

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Alessio Angius ◽  
András Horváth ◽  
Sami M. Halawani ◽  
Omar Barukab ◽  
Ab Rahman Ahmad ◽  
...  
Keyword(s):  
Probability Density ◽  
Padé Approximation ◽  
Moment Generating Function ◽  
Matrix Exponential ◽  
Stable Method ◽  
Pade Approximation ◽  
Exponential Distributions ◽  
Special Cases ◽  
The Moment ◽  
And Behavior

This paper deals with moment matching of matrix exponential (ME) distributions used to approximate general probability density functions (pdf). A simple and elegant approach to this problem is applying Padé approximation to the moment generating function of the ME distribution. This approach may, however, fail if the resulting ME function is not a proper probability density function; that is, it assumes negative values. As there is no known, numerically stable method to check the nonnegativity of general ME functions, the applicability of Padé approximation is limited to low-order ME distributions or special cases. In this paper, we show that the Padé approximation can be extended to capture the behavior of the original pdf around zero and this can help to avoid representations with negative values and to have a better approximation of the shape of the original pdf. We show that there exist cases when this extension leads to ME function whose nonnegativity can be verified, while the classical approach results in improper pdf. We apply the ME distributions resulting from the proposed approach in stochastic models and show that they can yield more accurate results.

Inequalities with application in economic risk analysis

1972 ◽  
Vol 9 (02) ◽  
pp. 441-444 ◽  
Author(s):  
Robert A. Agnew
Keyword(s):  
Risk Analysis ◽  
Lower Bounds ◽  
Generating Function ◽  
Random Variable ◽  
Moment Generating Function ◽  
Upper Bounds ◽  
Economic Risk ◽  
The Moment

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

scholarly journals A New Flexible Generalized Lindley Model: Properties, Estimation and Applications

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1678
Author(s):  
Abdulrahman Abouammoh ◽  
Mohamed Kayid
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Estimation ◽  
Real Data ◽  
Moment Generating Function ◽  
Rate Function ◽  
Failure Rate Function ◽  
Right Censored Data ◽  
Special Cases ◽  
The Moment ◽  
The Right

A new method for generalizing the Lindley distribution, by increasing the number of mixed models is presented formally. This generalized model, which is called the generalized Lindley of integer order, encompasses the exponential and the usual Lindley distributions as special cases when the order of the model is fixed to be one and two, respectively. The moments, the variance, the moment generating function, and the failure rate function of the initiated model are extracted. Estimation of the underlying parameters by the moment and the maximum likelihood methods are acquired. The maximum likelihood estimation for the right censored data has also been discussed. In a simulation running for various orders and censoring rates, efficiency of the maximum likelihood estimator has been explored. The introduced model has ultimately been fitted to two real data sets to emphasize its application.

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