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.
The following is a possible definition of the Beta function:
Definition The Beta function is a function defined as follows:where is the Gamma function.
While the domain of definition of the Beta function can be extended beyond the set of couples of strictly positive real numbers (for example to couples of complex numbers), the somewhat restrictive definition given above is more than sufficient to address all the problems involving the Beta function that are found in these lectures.
The Beta function has several integral representations, which are sometimes also used as a definition of the Beta function, in place of the definition we have given above. We report here two often used representations.
The first representation involves an integral from zero to infinity:
Given the definition of the Beta function as a ratio of Gamma functions (see above), the equality holds if and only if orThat the latter equality indeed holds is proved as follows:
Another representation involves an integral from zero to one:
This can be obtained from the previous integral representation:by performing a change of variable. The change of variable isBefore performing it, note thatand thatFurthermore, differentiating the previous expression we obtainWe are now ready to perform the change of variable:
Note that the two representations above involve improper integrals that converge if and : this might help you to see why the arguments of the Beta function are required to be strictly positive.
The following sections contain more details about the Beta function.
The integral representation of the Beta functioncan be generalized by substituting the upper bound of integration () with a variable ():The function thus obtained is called incomplete Beta function.
Below you can find some exercises with explained solutions.
Compute the following product:where is the Gamma function and is the Beta function.
We need to write the Beta function in terms of Gamma functions:where we have used several elementary facts about the Gamma function, that are explained in the lecture entitled Gamma function.
Compute the following ratiowhere is the Beta function.
This is achieved by rewriting the numerator of the ratio in terms of Gamma functions and using the recursive formula for the Gamma function:
Compute the following integral:
We need to use the integral representation of the Beta function:Now, write the Beta function in terms of Gamma functions:Substituting this number into the previous expression for the integral, we obtainIf you wish, you can check the above result by using the following MATLAB commands:
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