For questions about the Beta function (also known as Euler's integral of the first kind), which is important in calculus and analysis due to its close connection to the Gamma function. It is advisable to also use the [special-functions] tag in conjunction with this tag.

The Beta function is a function of two variables that is often found in probability theory and mathematical statistics (for example, as a normalizing constant in the probability density functions of the $F$ distribution and of the Student's $t$ distribution). We report here some basic facts about the Beta function.

Definition: The Beta function, denoted by $B(x,y)$, is defined as $$B(x,y)=\int_0^1 t^{x-1}~(1-t)^{y-1}~dt$$ This is also the Euler's integral of the first kind.

Relation between Beta function and Gamma function: $$B(x,y)=\frac{\Gamma(x)~\Gamma(y)}{\Gamma(x+y)}$$ For positive integers $~x~$ and $~y~$, we can define the beta function as $$B(x,y)=\frac{(x-1)!~(y-1)!}{(x+y-1)!}$$

Application:

Beta function is widely applicable. It is utilized in various fields, few of them are described below:

$1)~$ Beta functions are commonly used in probability theory. It is a part of the family of continuous probability distributions.

$2)~$ Beta functions may be used for statistical description in population genetics.

$3)~$ This function is quite frequently used in differential calculus as well as in integral calculus.

$4)~$ Not only in mathematics, beta functions are utilized in other areas too such as - physics, engineering and technology.

References:

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

http://mathworld.wolfram.com/BetaFunction.html