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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.
The Beta distribution on [0,1], a family of two-parameter distributions with one mode, of which the uniform distribution is a special case, and which is useful in estimating success probabilities. The four-parameter Beta distribution, a straight-forward generalization of the Beta distribution to arbitrary bounded intervals [,].
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:
Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative: ¯ = (), or denoted as (), In particular, the pdf of the standard normal distribution is denoted by φ ( z ) {\textstyle \varphi (z)} , and its cdf by Φ ( z ) {\textstyle \Phi (z)} .
The beta negative binomial distribution contains the beta geometric distribution as a special case when either = or =. It can therefore approximate the geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large α {\displaystyle \alpha } .
Some distributions have been specially named as compounds: beta-binomial distribution, Beta negative binomial distribution, gamma-normal distribution. Examples: If X is a Binomial(n,p) random variable, and parameter p is a random variable with beta(α, β) distribution, then X is distributed as a Beta-Binomial(α,β,n).
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The geometric distribution is the only memoryless discrete probability distribution. [4] It is the discrete version of the same property found in the exponential distribution . [ 1 ] : 228 The property asserts that the number of previously failed trials does not affect the number of future trials needed for a success.