Beta Distribution

–  By Niki Gandhi

The Beta distribution is defined as a probability distribution which is continuous. It has two positive shape parameters denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. A family of probabilities is represented by this distribution and is a versatile way to represent outcomes for percentages or proportions. For example, how likely is it that BJP will win Gujarat 2017 elections? You might think the probability is 0.2. Your friend might think it’s 0.15. The beta distribution gives us a way to describe this.

The word “beta” has 3 completely different meanings.

1. Beta (α, β) where “Beta” is the name of the probability distribution.
2. B (α, β) where “Beta” is the name of a function that appears in the denominator of the density function.
3. β – Here, “Beta” is the name of the second parameter in the density function.

Characterization of Beta Distribution:

Definition: Let a variable X be an absolutely continuous random variable and its support be the unit interval:

Rx = [0,1]

Let  α,β be a subset of R. We can say that variable X has a Beta distribution with the shape parameters α and β if its probability density function is

where B() is the Beta function.

Beta random variable is a random variable having a Beta distribution.

Probability Density Function

The probability density function of the beta distribution is represented as a power function of the variable x and of its reflection (1 − x), for 0 ≤ x ≤ 1, and shape parameters α, β > 0.

Properties of Beta Distribution

Measures of central tendency of a Beta distribution are

1. Mode of Beta Distribution: The modeof a Beta distributed random variable X with α, β > 1 is the most likely value of the distribution. Anti-mode is the lowest point of the probability density curve, when both parameters are less than one (α, β < 1).
2. Median of Beta Distribution: The median of the beta distribution is the unique real number. {\displaystyle x=I_{\frac {1}{2}}^{[-1]}(\alpha ,\beta )}{\displaystyle I_{x}(\alpha ,\beta )={\tfrac {1}{2}}}{\displaystyle {\frac {\alpha -1}{\alpha +\beta -2}}.}
3. Mean of Beta Distribution: The expected value (mean) of a Beta distribution of random variable Xwith two parameters α and β is a function of only the ratio β/α of these parameters.

The total area under the density curve equals 1 which is assured by the Beta function B in the denominator also known as the “normalizing constant”. The Beta function is equal to a ratio of Gamma functions:

B(α, β) = Γ(α)*Γ(β)/ Γ(α+β)

The expected value of a variable that is Beta distributed is: E(x) = µ = α / α+β and the variance is given by Variance(x) = [ β / (α+β) (α+β + 1) ] * µ

Use of Beta Distribution:

It is used to model one’s uncertainty about the probability of success of an experiment.

References:

https://en.wikipedia.org/wiki/Beta_distribution

https://www.statlect.com/probability-distributions/beta-distribution

http://pj.freefaculty.org/guides/stat/Distributions/DistributionWriteups/Beta/Beta.pdf

http://www.statisticshowto.com/beta-distribution/