Gamma Distribution

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  eBook - PDF

  Reinforced Concrete Structural Reliability

  • Ph.D, Mohamed Abdallah El-Reedy(Authors)
  • 2012(Publication Date)
  • CRC Press

    (Publisher)

  This distribu-tion is complicated and, therefore, is not recommended for use in the case of building a huge model of an entire problem when using the Monte Carlo simulation. 2.3.7 Gamma Distribution Gamma Distribution (Figure 2.13) represents a large number of events and transactions, such as inventory control or representation of economic Probability Density 0 0.5 0.1 2.0 1.5 x 2.5 3.0 β = 0.5 β = 1 β = 1.5 β = 3.2 β = 5 FIGURE 2.12 Weibull distribution. 38 Reinforced Concrete Structural Reliability theories. The theory of risk insurance also is used in environmental studies when there is a concentration of pollution. It is used in modeling the live load in the stochastic study, which is discussed in Chapter 4. On the other hand, this distribution is applied to model the time to the arrival of a fixed number of vehicles, or the time to the failure of a system designed to accept a fixed number of overloads before falling when the vehicles and the overloads are Poisson distributions. In fact, the Gamma Distribution is more broadly defined than is implied by its derivation as the distribution of the sum of k independent of iden-tically distributed exponential random variables. More generally, the parameter K need not be integer-valued when the Gamma Distribution is written as: fx x x c k k hx ( ) ( ) = ( ) --λ λ 1 Γ x ≥0 (2.25) The only restriction being λ > 0. The gamma function Γ ( k ) (from which the distribution gets its name) is equal to ( k – 1) if k is an integer, but more gener-ally is defined by the definite integral: Γ ( ) k e u d u u k = --∞ ∫ 1 0 (2.26) 0.5 0.4 k = 1, θ = 2.0 k = 2, θ = 2.0 k = 3, θ = 2.0 k = 5, θ = 1.0 k = 9, θ = 0.5 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 16 18 20 FIGURE 2.13 Gamma Distribution. 39 Main Statistics in Structure Engineering The integral arises here as a constant, which is needed to normalize the function. We say x is gamma-distributed with parameters k and λ , G(k, λ ), if its density function has the form in equation (2.23).

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  Simple Statistical Methods for Software Engineering

  Data and Patterns

  • C. Ravindranath Pandian, Murali Kumar(Authors)
  • 2015(Publication Date)
  • Auerbach Publications

    (Publisher)

  Applying Gamma Distribution to Software Reliability Growth Modeling Gamma Distribution has been used in reliability studies. The gamma cumulative dis-tribution function (CDF) is an S curve and can represent cumulative defects discov-ered in software. An example of CDF, although on a different metric, may be found in Figure 18.5. A CDF plotted with cumulative defects is known as the software reliability growth curve, a subject treated in detail in Chapter 19 where the Weibull distribution is used and in Chapter 21 where the Gompertz distribution is used. Although the Weibull distribution is popularly used to fit failure data, the Gamma Distribution is more suited in certain cases. In a research study, Sonia Meskini [6] applies Gamma Distribution to construct a software reliability growth model for smart phones. Sonia observes, Working with a PDF such as gamma has further advantages. First, gaps in data are filled by the equation. Second, one can extrapolate mathematically and see beyond boundaries. Third, strategic planners can estimate risk. There are several options available while choosing a PDF for rainfall data. Statistical techniques include the compound Poisson–exponential distribu-tion; the log, square root, and cube root normal distributions; the Gamma Distribution; various normalizing transforms; the kappa distribution, the Weibull distribution, and the Box–Cox transformation. However, the Gamma Distribution is frequently used to represent precipita-tion because it provides a flexible representation of a variety of distribution shapes while using only two parameters: shape and scale (based on Husak [4]). 296 ◾ Simple Statistical Methods for Software Engineering In Skype Version I the Weibull distribution is the closest to the actual behavior curve of the application, followed by the Gamma Distribution.

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  The Probability Handbook

  • Mary McShane-Vaughn(Author)
  • 2016(Publication Date)
  • ASQ Quality Press

    (Publisher)

  • A production engineer uses the Gamma Distribution to model the time between machine breakdowns. • A reliability engineer uses the Gamma Distribution to estimate the mean time between unit failures. • An engineer at an amusement park uses the Gamma Distribution to model guests’ time in queue for the new roller coaster. Figure 5.39 Microsoft Excel function =gamma.dist(). Figure 5.40 Microsoft Excel function =gamma.inv(). CONTINUOUS PROBABILITY DISTRIBUTIONS 121 5.9 CHI-SQUARE DISTRIBUTION The chi-square (pronounced kai-square) sampling distribution is a workhorse in sta- tistical analysis. It is used to perform parametric tests on the variance of a normal distribution, and in nonparametric tests of independence for contingency tables, goodness-of-fit tests, and many specialty applications such as the Mantel-Haenszel test for independence measured over time. 6 5.9.1 Probability Density Function The chi-square distribution is a special case of the Gamma Distribution with α = r/2 and β = 2. The term r is the degrees of freedom for the chi-square distribu- tion. The chi-square random variable X is a function of the sample variance of the normal distribution, s 2 , as shown in Formula 5.33. The term (n – 1)s 2 /σ 2 follows a chi-square distribution with r = (n – 1) degrees of freedom: Formula 5.33 Chi-square random variable ~ χ 2 (n – 1) (n – 1)s 2 σ 2 The probability density function of the chi-square distribution is shown in For- mula 5.34. Formula 5.34 Probability density of chi-square distribution f(x) = , x > 0 Γ 2 r 2 x e –x/2 r 2 ) ) ) ( 2 – 1 r ) ( The expected value and variance of the chi-square distribution are displayed in Formulas 5.35 and 5.36, respectively. Formula 5.35 Expected value of chi-square distribution E(X) = r Formula 5.36 Variance of chi-square distribution Var(X) = 2r 6 Refer to http://www.biostathandbook.com/cmh.html for an interesting explanation of the Mantel-Haenszel test involving ponies and pink leg warmers.

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  Probability Theory and Statistical Applications

  A Profound Treatise for Self-Study

  • Peter Zörnig(Author)
  • 2016(Publication Date)
  • De Gruyter

    (Publisher)

  As the name suggests, this model is based on the gamma function (see (1.3)). Definition 10.3: A continuous random variable X assuming all values x ≥ 0 is said to have a Gamma Distribution , if its pdf is given by f x α r αx e x ( ) = Γ( ) ( ) for > 0. r a x -1 -(10.11) The distribution depends on the two positive parameters α and r . The first of them is a scaling parameter and the latter characterizes the shape of the distribution. Using the definition of the gamma function and the substitution y = αx it follows that ∫ ∫ ∫ f x dx αx e αdx y e dy ( ) = ( ) = = 1 r r α x r r y 0 ∞ 1 Γ( ) 0 ∞ -1 -1 Γ( ) 0 ∞ -1 - 140 10 Continuous probability models r = 4 r = 2 r = 1/2 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 f(x) 0 1 2 3 x 4 5 6 7 Fig. 10.4 : Densities of the gamma family The pdf is illustrated in Fig. 10.4 for α = 1 and some values of r . The gamma distribu-tion reduces to the exponential distribution, setting r = 1 in (10.11). Theorem 10.4: If X has the Gamma Distribution (10.11) then (i) E ( X ) = r α (ii) V ( X ) = r α 2 Example 10.4: The life time X of an electronic component has a Gamma Distribution with r = 2. If the life expectancy is 50 hours, what is the probability that the compo-nent functions at least 60 hours? Solution: Using Theorem 10.4(i) and (10.11) we obtain α = 2/50 = 0.04 and the pdf is f ( x ) = α x e a x 2 -Hence, integration by parts yields ∫ P X α xe dx ( ≥ 60 = ≈ 0.3084 α x 60 ∞ 2 -Example 10.5: Assume that the time to failure (in hours) of an antiaircraft missile is a gamma-distributed random variable X . 10.4 The chi-square distribution 141 (i) What is the probability that the system does not fail during a period of 24 hours, if the parameters are α = 0.1 and r = 3.2.

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  System Reliability Theory

  Models, Statistical Methods, and Applications

  • Marvin Rausand, Anne Barros, Arnljot Hoyland(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  In spite of this limited usage, the Gamma Distribution is an important distribution in reliability because it is used in other situations as illustrated later in this book (e.g. see Chapter 15). The gamma function is available in R by the command gamma(x), for example, gamma(2.7)= 1.544686. In R, the parameter  is called shape and  is called rate. We may alternatively use the parameter  = 1∕, which is called scale in R, as the second parameter. The probability density functions (e.g. for  = 2 and  = 1 can be plotted by the R script: t<-seq(0,6,length=300) # Set time axis # Set the parameters a<-2 # shape rate<-1 # rate # Calculate the gamma density (y) for each t y<-dgamma(t,a,rate,log=F) plot(t,y,type="l") Observe that we have to write rate= to specify  in the script. We could, alterna- tively, have written scale= to specify the scale parameter (= 1∕). The scale parameter is the default parameter in R and if we write only the number, it is inter- preted as scale. From (5.57) we find that MTTF =   = . (5.58) var(T) =   2 =  2 . (5.59) 170 5 Probability Distributions in Reliability Analysis The parameter  is a dimensionless number, whereas  is measured in time units (e.g. hours). For a specified value of , the MTTF is proportional to . The distribution function F(t) is available in R by the command pgamma and R(t) is than obtained as 1-pgamma. The survivor function R(t) (e.g. for  = 2 and  = 1) can be plotted by the R script: t<-seq(0,6,length=300) # Set time axis # Set the parameter a<-2 # shape rate <- 1 #rate # Calculate the survivor function (y) for each t y<-1-pgamma(t,a,rate,log=F) plot(t,y,type="l") A sketch of R(t) is given in Figure 5.18 for some values of .

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  Generalized Frequency Distributions for Environmental and Water Engineering

  • Vijay P. Singh, Lan Zhang(Authors)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  6 Three-Parameter Generalized Gamma Distribution 6.1 Introduction Two-parameter Gamma Distribution is widely applied in environmental and water engineering. However, it has limited flexibility and versatility. Hence, its generalizations as three-parameter and four-parameter Gamma Distributions have been proposed and applied. The generalized gamma (GG) distributions have been employed for modeling flood peaks, volumes, and mean values. The Gamma Distribution has also been used in rainfall-runoff modeling (see Lienhard, 1964, 1972; Lienhard and Meyer 1967; Singh, 1988, 2013). In Russia, China, and Eastern Europe, the three-parameter generalized gamma (TPGG) distribution is also referred to as the Kritsky–Menkel distribution (Klemes, 1989). Bobée and Ashkar (1991) discussed the gamma family and its applications in hydrology. This chapter derives the TPGG distribution using the entropy theory and derives its parameters using this theory (Singh, 1998) as well as methods of maximum likelihood estimation (MLE) (Natural Environment Research Council, 1999), moments (Chebana et al., 2010), prob- ability weighted moments (Landwehr et al., 1979), L-moments (Hosking, 1990), and entropy method (Papalexiou and Koutsoyiannis, 2012). The distri- bution is also applied to real data. 6.2 Characteristics of GG Distribution Let X be a random variable and x be its specific value. The probability density function (PDF) of the TPGG distribution of X can be expressed as 162 f ðxÞ ¼ c bΓ a c   x b   a1 exp   x b   c  ; a, b, c > 0, x > 0; (6.1) where Γ() is the gamma function, a and c are the shape parameters, and b is the scale parameter. Figure 6.1 shows the tree diagram of the GG and its special distributions and Figure 6.2 shows that the GG distribution can take on different shapes for different values of parameters. c = 1 GG LN GA Weibull EXP a = c = 1 a = c Figure 6.1 Tree diagram of the GG and its special distributions.

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  PROBABILITY AND STATISTICS FOR ENGINEERS AND SCIEN

  • Anthony Hayter(Author)
  • 2012(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4.3 THE Gamma Distribution 201 FIGURE 4.13 Probability density functions of Gamma Distributions 0.5 1.0 2.5 5.0 7.5 10.0 f ( x ) x λ = 1, k = 3 λ = 1, k = 5 λ = 1, k = 1 FIGURE 4.14 Probability density functions of Gamma Distributions 0.5 1.0 2.5 5.0 7.5 10.0 f ( x ) x λ = 2, k = 3 λ = 1, k = 3 λ = 3, k = 3 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 202 CHAPTER 4 CONTINUOUS PROBABILITY DISTRIBUTIONS FIGURE 4.15 Distance to fifth fracture has a Gamma Distribution with parameters k = 5 and λ = 4 . 3 X X ~ Gamma k = 5, λ = 4.3 4.3.2 Examples of the Gamma Distribution Example 32 Steel Girder Fractures Suppose that the random variable X measures the length between one end of a girder and the fifth fracture along the girder, as shown in Figure 4.15. If the fracture locations are modeled by a Poisson process as discussed previously, X has a Gamma Distribution with parameters k = 5 and λ = 4 . 3. The expected distance to the fifth fracture is therefore E ( X ) = k λ = 5 4 . 3 = 1 . 16 m A software package can be used to show that the 0.05 quantile point of this distribution is x = 0 .

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