**Beta Distribution**

Table of Content

Introduction to Beta Distribution

Definition of Beta Distribution

Kinds of Beta Distribution

Beta Distribution of Different Values of α and β

Some Important Properties of the Beta Distribution

Measures of Dispersion of Beta Distribution

Shapes of Beta Distribution at Different Values of α and β

The relation between the Beta first kind and Beta second kind Distribution

Applications of Beta Distribution

Derivation of Beta Distribution from Other Distributions

Example of Beta Distribution

Conclusions of Beta Distribution

Introduction to Beta Distribution

Beta Distribution is one of the important distributions in the probability distribution family.

It is used to specify or relate to one or more random variables which take values within a

certain range having an upper and a lower limit. It is denoted by two shape parameters α and

β.

For example, if we take a set of articles with a certain proportion of defect in it.

Now we want to calculate the probability that certain percentage of the lot is found to

be defective. In these types of cases, we use Beta Distribution to calculate the desired

probabilities.**Definition of Beta Distribution**

Definition: A family of probability densities of continuous random variables whose

values are in the interval (0, 1) or (0, ∞) is defined as the family of Beta Distribution.

However, Beta Distribution and Beta Functions should not be treated as the same thing. Beta

Distribution is developed on the structural basis of Beta function, but they have some

differences as well.

Kinds of Beta Distribution

Two kinds of Beta functions are available in mathematics.

Let us discuss these two formats in detail:

(a) First kind: This format of the distribution is most widely used.The Beta Function of the

first kind is defined as:

and this is evaluated as

where

which is called the Gamma Function in Mathematics.

The well- known Gamma Distribution is developed on the basis of this function.

Based on this function, the Probability Density Function (PDF) of Beta first kind is given

by:

The random variable X is said to follow a Beta Distribution of the first kind if its PDF is

given by:

the family of Beta densities is a two-parameter family of densities that is positive on the

interval (0, 1).

Median of Beta Distribution

Mode of Beta Distribution: Mode or peak of the distribution is at

Measures of Dispersion of Beta Distribution

The variance of Beta Distribution

Proof of Variance:

(by the integral representation of the Beat Function)

(by the definition of Beta Function)

the shape of Beta Distribution

the shape of a distribution is usually described using two characteristics: (1) Skewness, and (2)

Kurtosis. Skewness is the measure of the asymmetry of the probability distribution of a random

variable about its mean. Kurtosis is the sharpness of the peak of a frequency- distribution

curve. Beta Distribution reduces the Rectangular Distribution or Continuous Uniform

Distribution over the interval (0, 1) if α = β = 1.

This Distribution pattern varies according to the variations of different shapes depending on

different values of the shape parameters.

Skewness of Beta Distribution

If α = β >1 then the distribution is symmetric.

If α < β, the distribution is positively skewed.

If α > β, the distribution is negatively skewed.

Kurtosis of Beta Distribution

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