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).
Beta Distribution of Different Values of α and β
The Cumulative Distribution Function (CDF), Moment Generating Function (MGF) of
this distribution is as follows:
CDF of Beta Distribution
where
This can be evaluated using the tables derived for Incomplete Beta Distribution.
MGF of Beta Distribution:
although, the MGF cannot be expressed in a compact form, the moments of the distribution
can be calculated using this by putting different method values of r.
Some Important Properties of the Beta Distribution
Measures of Central Tendency of Beta Distribution
Arithmetic Mean of Beta Distribution
Proof of Expectation
Harmonic Mean of Beta Distribution
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
Kurtosis or the peak of this distribution also depends on the value of the shape parameters.
For different values of the shape parameters, different peaks for the values are observed.
Shapes of Beta Distribution at Different Values of α and β
(b) Second kind: The Beta function of the second kind is defined as:
Which is evaluated using the expression stated above.
Based on this function the Probability Density Function (PDF) of Beta distribution of the second
kind is given by:
The random variable X is said to follow a Beta Distribution of the second kind if its PDF is given
by:
If 0< x< ∞, α >0 and β > 0 = 0 , otherwise
the Cumulative Distribution Function (CDF), mean and variance of this distribution areas
follows:
CDF of Beta Distribution
This can be evaluated using the tabulated values derived from Incomplete Beta Distribution.
The order raw moment of Beta Distribution is given by
Mean of Beta Distribution (Second kind)
The variance of Beta Distribution (Second kind)
Harmonic Mean of Beta Distribution (Second kind)
The MGF for Beta second kind does not exist as the integral is not absolutely
convergent.For the second kind as well, the shape of the Distribution depends on the
value of the shape parameters.
The relation between the Beta first kind and Beta second kind Distribution
If X ~ B (α, β) of the second kind, then follows Beta First kind with parameters
(α, β).
Applications of Beta Distribution
Beta Distribution is used for many purposes:
The beta distribution is used in describing the distribution of kth-order Statistics of the
Continuous Uniform Distribution.
Rule of Succession is another equally important and vital application of Beta
Distribution. By this rule the Posterior Distribution of p, the success probability is
characterized as Beta.
It also has immense importance in Bayesian Inference in accordance with the fact
that it provides the family of conjugate prior Probability Distributions namely
Binomial (as well as Bernoulli ) and Geometric.
In subjective logic, i.e. in areas where testing is required as to whether a proposition
about the real world is true or false, Beta distribution provides the posterior
probability estimates of binary events, i.e., to say, it is based on the reasoning from
the priory known facts rather than being based on assumptions.
Task cost and Schedule modeling are done with the help of Beta Distribution
Derivation of Beta Distribution from Other Distributions
The kth order statistics of a sample of size n from the uniform distribution is a beta
random variable, U(k)~ Beta(k, n+1−k).
If X ~ Gamma(α, θ) and Y ~ Gamma(β, θ) are independent, then it is said to follow
Beta (α, β) distribution.
If independently, follows Beta (α/2, β/2)
distribution.
If X ~ U(0, 1) and α > 0 then ~ Beta(α, 1). The power function distribution.
Example of Beta Distribution
Batting 0.400: 13 times from 1900 -1941, never since. Find the parameters of the beta
distribution so that μ = 0.260 and σ = 0.04. With these values of α and ß, what is the
probability a batting average exceeds 0.400?
α = 31.005, ß = 88.245
1-beta (0.4, 31.005, 88.245)
Consider that we believe exam scores for a particular class follow a beta distribution with α =
5 and ß = 2. We are interested in finding the answer to the following questions.
1. What is the expected score on the exam?
2. If 90% is considered an A, what proportion of students received an A?
3. If 60% is considered failing, what percentage of students failed? What percentage passed?
4. What is the standard deviation of exam scores?
Let X be the random variable denoting the exam scores of a particular class.
X~ Beta(5,2)
Then E(X) = =0.7142.
V(X) =
SD(X) = 0.15972.
P[X ≥ 0.9] =
P[X ≤ 0.6] =
Conclusions of Beta Distribution
The beta distribution is an important continuous distribution having various implications which
are dependent on parameters such as skewness and kurtosis. All important properties and its
uses are explained in this content.
