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In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
Random variables are usually written in upper case Roman letters, such as or and so on. Random variables, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable.
Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities.
Harmonic mean for Beta distribution for 0 < α < 5 and 0 < β < 5 (Mean - HarmonicMean) for Beta distribution versus alpha and beta from 0 to 2 Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), smaller values alpha and beta in front Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), larger values alpha and beta in front
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
The mathematics of the distribution resulted from the authors' desire to make the standard deviation equal to about 1/6 of the range. [ 2 ] [ 3 ] The PERT distribution is widely used in risk analysis [ 4 ] to represent the uncertainty of the value of some quantity where one is relying on subjective estimates, because the three parameters ...
In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind [1]) is an absolutely continuous probability distribution. If [,] has a beta distribution, then the odds has a beta prime distribution.
And in fact this is a special case of a more general theorem (See p. 59 of [18]) which allows any symmetric alpha-stable distribution to be viewed in this way (with the alpha parameter of the mixture distribution equal to twice the alpha parameter of the mixing distribution—and the beta parameter of the mixing distribution always equal to one).