beta distribution

beta distribution, continuous probability distribution used to represent outcomes of random behavior within fixed bounds, usually the range from 0 to 1. Beta distributions have two parameters α and β, which define the distribution’s shape in a range [a,b], where a is less than b. The lower limit, a, is called the location, and the range (b − a) is called the scale.

The values of α and β are typically calculated on the basis of the random variable’s trend within the fixed bounds, and they are analogous to the odds of success and failure, respectively. When a = 0 and b = 1, the beta distribution is known as the standard beta distribution. The beta distribution is often used to model the probability of success of a random experiment within the range [0,1].

Mathematical formula

The formula to calculate the probability density function (PDF), or simply probability, of the beta distribution isf(x) = (x −  a)^{(α − 1)}(b −  x)^{(β − 1)}/B(α,β) ∙ (b − a)^{(α + β − 1)},

where a ≤ x ≤ b, α and β > 0, and B (α,β) is a normalizing constant used to ensure that the area under the curve of the PDF equals 1. It is calculated byB(α,β) = Integral on the interval [0, 1 ] of ∫ 0 1 t^{(α − 1)}(1 − t)^{(β − 1)}dt

The formula for the PDF of the standard beta distribution, where a = 0 and b = 1, simplifies tof(x) = x^{(α − 1)}(1  −  x)^{(β − 1)}/B(α,β),

where 0 ≤ x ≤ 1 and α and β > 0. The average or expected value of the standard beta distribution isE(x) = α/(α + β)

The variance of the standard beta distribution isVar(x) = αβ/(α + β)²(α  +  β + 1)

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As α increases, the distribution shifts to the right. An increase in β shifts the curve to the left. Large values for both α and β narrow the curve, indicating a higher level of certainty in determining the outcome. If α and β are both smaller than 1, the distribution will take on an inverted normal or U-shape, with the chances of success highest at the extremes.

Similar distributions

When α and β both equal 1, the beta distribution simplifies to a uniform distribution, with the value of 1 within the range [0,1], and 0 otherwise. When α and β are sufficiently large, and approximately equal to each other, the beta distribution takes on the shape of a normal distribution.

The beta distribution is closely related to the binomial distribution. Whereas the binomial distribution models the number of successes in a given number of binary trials, the beta distribution can model the likelihood of success in these trials and helps determine the certainty (or uncertainty) of success. The beta distribution can thus be seen as the continuous variation of the binomial distribution.