page_title: Beta distribution
section_title: Mode
hierarchical_section_title: Beta distribution / Properties / Measures of central tendency / Mode
caption_attribution_description: English: Mode Beta Distribution for alpha and beta from 1 to 5
context_page_description: In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.
  The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines.
  In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. The beta distribution is a suitable model for the random behavior of percentages and proportions.
  The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas beta distribution of the second kind is an alternative name for the beta prime distribution.
context_section_description: The mode of a Beta distributed random variable X with α, β > 1 is the most likely value of the distribution (corresponding to the peak in the PDF), and is given by the following expression:
  When both parameters are less than one (α, β < 1), this is the anti-mode: the lowest point of the probability density curve.
  Letting α = β, the expression for the mode simplifies to 1/2, showing that for α = β > 1 the mode (resp. anti-mode when α, β < 1), is at the center of the distribution: it is symmetric in those cases.  See Shapes section in this article for a full list of mode cases, for arbitrary values of α and β. For several of these cases, the maximum value of the density function occurs at one or both ends.  In some cases the (maximum) value of the density function occurring at the end is finite. For example, in the case of α = 2, β = 1 (or α = 1, β = 2), the density function becomes a right-triangle distribution which is finite at both ends. In several other cases there is a singularity at one end, where the value of the density function approaches infinity. For example, in the case α = β = 1/2, the Beta distribution simplifies to become the arcsine distribution.  There is debate among mathematicians about some of these cases and whether the ends (x = 0, and x = 1) can be called modes or not.
  Whether the ends are part of the domain of the density function
  Whether a singularity can ever be called a mode
  Whether cases with two maxima should be called bimodal
caption: Mode for Beta distribution for 1 ≤ α ≤ 5 and 1 ≤ β ≤ 5
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