# Distributions

• James J. Palestro
• Per B. Sederberg
• 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}

© Springer International Publishing AG 2018

## Authors and Affiliations

• James J. Palestro
• 1
• Per B. Sederberg
• 1