# 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

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 behavior of proportions or percentage.

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

Beta distribution is defined on the interval[0,1] denoted by $\alpha$ and $\beta$, usually. $\alpha$ and $\beta$ are two positive parameters that appears as exponents of the random variable and is intended to control the shape of distribution.

Its notation is Beta($\alpha$,$\beta$), where $\alpha$ and $\beta$ are the real numbers and the values are more than zero.

## Beta Distribution Formula

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

Beta distribution formula is basically defined based on Probability Density Function, Cumulative Density Function, Moment generating function, Expectation and Variance.

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

Also Check: Probability Density Function Calculator

The formula for Cumulative density function(CDF) is;

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

Where

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

When Beta is considered as the moment generating function then,

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

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

### 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

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

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#### Practise This Question

A person took some money with some interest for 1 year but later he found that the rate of interest was 2% extra. According to the new rate of interest, he paid Rs.60/- extra. How much money did he borrow?