Beta Distribution

The Beta distribution is a type of probability distribution which represents all the possible value of probability. Let us discuss, its definition and formula with examples.

Beta Distribution Definition

In probability and statistics, the Beta distribution is considered as a continuous probability distribution defined by two positive parameters. It is a type of probability distribution which is used to represent the outcomes or random behaviour of proportions or percentage. The most common use of this distribution is to model the uncertainty about the probability of success of a random experiment.

Usually, the basic distribution is known as the Beta distribution of its first kind and beta prime distribution is called for its second kind.

Beta Distribution

It is defined on the interval [0,1] denoted by α and β, usually. α and β are two positive parameters that appear as exponents of the random variable and is intended to control the shape of the distribution. Its notation is Beta(α,β), where α and β are the real numbers and the values are more than zero.

Beta Distribution Formula

The beta distribution is used to check the behaviour of random variables which are limited to intervals of finite length in a wide variety of disciplines.

The characterization of this distribution is basically defined as Probability Density Function, Cumulative Density Function, Moment generating function, Expectations and Variance and its formulas are given below.

Probability Density Function

The formula for the probability density function(PDF) is;

PDF=\(x^{a-1}(1-x)^{\beta -1}/B(\alpha ,\beta )\)

B(\alpha ,\beta )\)=\(\Gamma \alpha +\Gamma \beta /\Gamma \alpha \Gamma \beta\)

or

B(\alpha ,\beta )\)=\(\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt\)

Where \(\Gamma(k)!=(k-1)!=1 \times 2 \times\ 3 \times\cdots \times (k-1)\)

Cumulative Density Function

The formula for Cumulative density function(CDF) is;

CDF=\(B_{x}(\alpha ,\beta )/B(\alpha ,\beta )\)

Where

\(B_x(\alpha,\beta)=\int_0^x t^{\alpha-1}(1-t)^{\beta-1}dt\) \(B(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt\)

Moment Generating Function

When Beta is considered as the moment generating function then,

M(t)=\(1+\sum_{k=1}^\infty (\prod_{r=0}^{k-1}\frac{\alpha+r}{\alpha+\beta+r})\frac{t^k}{k!}\)

Expectations

The expected value of Beta distributed random variable say x is;

E(X)=\(\frac{\alpha}{\alpha+\beta}\)

The Variance value of Beta or Beta Variance is;

Var(X)=\(\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\)

Properties

Some of the properties that  satisfy the distribution are as follow:

The measure of central tendency

  1. Mean
  2. Median
  3. Mode
  4. Geometric Mean
  5. Harmonic Mean

The measure of statistical dispersion

  1. Variance
  2. Geometric variance and co variance
  3. Mean absolute difference
  4. Mean absolute deviation around the mean

Beta Distribution Applications

It is used in many applications, that includes

  • Bayesian hypothesis testing
  • The rule of succession
  • Task duration modelling
  • Project planning control or management systems like CPM and PERT.

Beta Distribution Example

Problem: Suppose, if in a basket there are balls which are defective with a Beta distribution of \(\alpha\)=5 and \(\beta\)=3 and the random value of variable x are 0.4. Compute the probability of defective balls in the basket.

Solution: Let us consider the balls are defective with a Beta distribution of \(\alpha\)=2 and \(\beta\)=5. Now to calculate the probability of defective balls from 20% to 30% in the basket we have to apply the Beta probability density function formula, which is;

P(x) = \(x^{a-1}(1-x)^{\beta -1}/B(\alpha ,\beta )\)

P(0.2\(\leq\)x\(\leq\)0.3)= \(\sum_{0.2}^{0.3}x^{2-1}(1-x)^{5 -1}/B(2 ,5 )\)

=0.235185

Beta Distribution Example

We hope with this example problem, the concept of beta distribution is understood.

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Practise This Question

Each of the letters in the word RIVAL is on a separate card, face down on the table. If you pick a card at random, what is the probability that its letter will be R?

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