by Marco Taboga, PhD
The Gamma function is a generalization of the factorial function to non-integer numbers.
It is often used in probability and statistics, as it shows up in the normalizing constants of important probability distributions such as the Chi-square and the Gamma.
In this lecture we define the Gamma function, we present and prove some of its properties, and we discuss how to calculate its values.
Table of contents
Introduction and motivation
Definition
How to compute the values of the function
Properties and caveats
Recursive formula
Relation to the factorial function
Some known values
Lower and upper incomplete Gamma functions
Solved exercises
Exercise 1
Exercise 2
Exercise 3
References
Introduction and motivation
Recall that, if , its factorial isso that satisfies the following recursion:
The Gamma function satisfies a similar recursion:but it is defined also when is not an integer.
Definition
The following is a possible definition of the Gamma function.
Definition The Gamma function is a function satisfying the following equation:
The domain of definition of the Gamma function can be extended beyond the set of strictly positive real numbers (for example to complex numbers).
However, the somewhat restrictive definition given above is sufficient to address the great majority of statistics problems that involve the Gamma function.
How to compute the values of the function
We will show below some special cases in which the value of the Gamma function can be derived analytically.
However, in general, it is not possible to express in terms of elementary functions for every .
As a consequence, one often needs to resort to numerical algorithms to compute .
We include here a calculator that implements one of these algorithms and we refer the reader to Abramowitz and Stegun (1965) for a thorough discussion of the main methods to compute numerical approximations of .
Plot of the gamma function with interactive calculator
Properties and caveats
If you play with the calculator, you will notice several properties of the Gamma function:
-
it tends to infinity as
approaches ; -
it quickly tends to infinity as
increases; -
for large values of
, is so large that an overflow occurs: the true value of is replaced by infinity; however, we are still able to correctly store the natural logarithm of in the computer memory.
The last point has great practical relevance. When we manipulate quantities that depend on a value taken by the Gamma function, we should always work with logarithms.
Recursive formula
Given the above definition, it is straightforward to prove that the Gamma function satisfies the following recursion:
Proof
The recursion can be derived by using integration by parts:
Relation to the factorial function
When the argument of the Gamma function is a natural number then its value is equal to the factorial of :
Proof
First of all, we have that
Using the recursion , we obtain
Some known values
A well-known formula, which is often used in probability theory and statistics, is the following:
Proof
By using the definition and performing a change of variable, we obtain
By using this fact and the recursion formula previously shown, it is immediate to prove that
for .
Proof
The result is obtained by iterating the recursion formula:
Lower and upper incomplete Gamma functions
The definition of the Gamma functioncan be generalized in two ways:
-
by substituting the upper bound of integration (
) with a variable (): -
by substituting the lower bound of integration with a variable:
The functions and thus obtained are called lower and upper incomplete Gamma functions.
Clearly, they have the property thatfor any , which is equivalent to
The two ratiosandare often called standardized incomplete Gamma functions.
They are numerically more stable and easier to deal with because they take values between and , while the values taken by the two functions and can easily overflow.
The lower incomplete function is particularly important in statistics, as it appears in the distribution function of the Chi-square and Gamma distributions.
Interactive calculator for lower and upper incomplete Gamma functions
Solved exercises
Below you can find some exercises with explained solutions.
Exercise 1
Compute the following ratio:
Solution
We need to repeatedly apply the recursive formulato the numerator of the ratio:
Exercise 2
Compute
Solution
We need to use the relation of the Gamma function to the factorial function: which, for , becomes
Exercise 3
Express the following integral in terms of the Gamma function:
Solution
This is accomplished as follows:
where in the last step we have just used the definition of Gamma function.
References
Abramowitz, M. and I. A. Stegun (1965) Handbook of mathematical functions: with formulas, graphs, and mathematical tables, Courier Dover Publications.
How to cite
Please cite as:
Taboga, Marco (2021). "Gamma function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/mathematical-tools/gamma-function.
