Gamma Probability Distribution Family

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A Gamma Probability Distribution Family is a two-parameter family of continuous probability distributions (of gamma density functions).



References

2022

2015

  1. p. 43, Philip J. Boland, Statistical and Probabilistic Methods in Actuarial Science, Chapman & Hall CRC 2007
  2. J. G. Robson and J. B. Troy, "Nature of the maintained discharge of Q, X, and Y retinal ganglion cells of the cat", J. Opt. Soc. Am. A 4, 2301–2307 (1987)
  3. M.C.M. Wright, I.M. Winter, J.J. Forster, S. Bleeck "Response to best-frequency tone bursts in the ventral cochlear nucleus is governed by ordered inter-spike interval statistics", Hearing Research 317 (2014)
  4. N. Friedman, L. Cai and X. S. Xie (2006) "Linking stochastic dynamics to population distribution: An analytical framework of gene expression", Phys. Rev. Lett. 97, 168302.
  5. DJ Reiss, MT Facciotti and NS Baliga (2008) "Model-based deconvolution of genome-wide DNA binding", Bioinformatics, 24, 396–403
  6. MA Mendoza-Parra, M Nowicka, W Van Gool, H Gronemeyer (2013) "Characterising ChIP-seq binding patterns by model-based peak shape deconvolution", BMC Genomics, 14:834
  7. See Hogg and Craig (1978, Remark 3.3.1) for an explicit motivation
  8. Scalable Recommendation with Poisson Factorization, Prem Gopalan, Jake M. Hofman, David Blei, arXiv.org 2014

2006

  • (Dubnicka, 2006g) ⇒ Suzanne R. Dubnicka. (2006). “Special Continuous Distributions - Handout 7." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
    • THE GAMMA FUNCTION: The gamma function is a function of t, defined for all k > 0 as ...
    • TERMINOLOGY : A random variable X is said to have a gamma distribution with parameters k > 0 and > 0 if its pdf is given by fX(x) =
      • 0, otherwise.
    • TERMINOLOGY : In the Gamma(k, ) family, when k = /2, for any integer , and = 1/2, we call the resulting distribution a 2 distribution with degrees of freedom. If X has a 2 distribution with degrees of freedom, we write X ∼ 2().

2004

  • (Arsham, 2004) ⇒ Hossein Arsham. (2004). “Topics in Statistical Data Analysis: Revealing Facts From Data." Webpage
    • Application: A basic distribution of statistics for variables bounded at one side - for example [math]\displaystyle{ x }[/math] greater than or equal to zero. Gives distribution of time required for exactly k independent events to occur, assuming events take place at a constant rate. Used frequently in queuing theory, reliability, and other industrial applications.
    • Example: Distribution of time between re-calibrations of instrument that needs re calibration after k uses; time between inventory restocking, time to failure for a system with standby components.
    • Comments: Erlangian, exponential, and chi-square distributions are special cases. The Dirichlet is a multidimensional extension of the Beta distribution.
    • Distribution of a product of iid uniform (0, 1) random? Like many problems with products, this becomes a familiar problem when turned into a problem about sums. If X is uniform (for simplicity of notation make it U(0,1)), Y=-log(X) is exponentially distributed, so the log of the product of X1, X2, … Xn is the sum of Y1, Y2, … Yn which has a gamma (scaled chi-square) distribution. Thus, it is a gamma density with shape parameter n and scale 1.