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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.
The uniform distribution or rectangular distribution on [a,b], where all points in a finite interval are equally likely, is a special case of the four-parameter Beta distribution. The Irwin–Hall distribution is the distribution of the sum of n independent random variables, each of which having the uniform distribution on [0,1].
¯ = sample mean of differences d 0 {\displaystyle d_{0}} = hypothesized population mean difference s d {\displaystyle s_{d}} = standard deviation of differences
The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents, [1] and the LaTeX symbol.
The beta family includes the beta of the first and second kind [7] (B1 and B2, where the B2 is also referred to as the Beta prime), which correspond to c = 0 and c = 1, respectively. Setting c = 0 {\displaystyle c=0} , b = 1 {\displaystyle b=1} yields the standard two-parameter beta distribution .
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The regularized incomplete beta function is the cumulative distribution function of the beta distribution, and is related to the cumulative distribution function (;,) of a random variable X following a binomial distribution with probability of single success p and number of Bernoulli trials n: