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Distributions

  • James J. Palestro
  • Per B. Sederberg
  • Adam F. Osth
  • Trisha Van Zandt
  • Brandon M. Turner
Chapter
Part of the Computational Approaches to Cognition and Perception book series (CACP)

Abstract

Here, we provide the PDFs for several distributions that we use throughout the book.

Here, we provide the PDFs for several distributions that we use throughout the book.

Beta Distribution

The probability of observing the random variable x under the Beta distribution with shape parameters α ∈ (0, ) and β ∈ (0, ) is
$$\displaystyle \begin{aligned} f(x|\alpha,\beta) = \dfrac{\varGamma(\alpha)\varGamma(\beta)}{\varGamma(\alpha+\beta)} x^{\alpha-1} (1-x)^{\beta-1} \end{aligned}$$
where Γ(x) = (x − 1)!.

Binomial Distribution

In n trials, the binomial distribution defines the probability of observing x = {0, 1, …, n} successes as
$$\displaystyle \begin{aligned} f(x|p,n) = {n \choose x} p^{x} (1-p)^{n-x}, \end{aligned}$$
where the probability of a single-trial success is the parameter \(p\in \left [ 0,1 \right ]\).

Gamma Distribution

The probability of observing the random variable x under the Gamma distribution with shape parameter k ∈ (0, ) and scale parameter θ ∈ (0, ) is
$$\displaystyle \begin{aligned} f(x|k,\theta) = \dfrac{1}{\varGamma(k) \theta^k} x^{k-1} \exp \left(-\dfrac{x}{\theta} \right). \end{aligned}$$

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • James J. Palestro
    • 1
  • Per B. Sederberg
    • 1
  • Adam F. Osth
    • 2
  • Trisha Van Zandt
    • 1
  • Brandon M. Turner
    • 1
  1. 1.Department of PsychologyThe Ohio State UniversityColumbusUSA
  2. 2.University of MelbourneParkvilleAustralia

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