The factorial of a natural number is the product of all natural numbers smaller than or equal to .
On this page we provide a basic introduction to factorials and we explain how they are used in probability theory and statistics.
The following is a formal definition.
Definition Let . The factorial of , denoted by , is:
The expression is read " factorial".
This definition is extended to the number 0 by using the convention:
For example, the factorial of 6 is
It is frequent to encounter ratios of factorials, which can be computed by simplifying the common terms. For example,
In the calculus of probabilities we often need to count permutations, combinations and partitions of objects. This can easily be done with factorials.
A permutation is one of the possible ways of ordering objects, from first to last.
The number of possible permutations is equal to .
Example Consider the first three letters of the alphabet: . There are ways of ordering these letters:
A combination is a way of selecting objects from a list of . The order of selection does not matter and each object can be selected only once.
The number of possible combinations is equal to
Example The number of possible ways to choose a team of three people from a group of five is
A partition is a way of subdividing objects into groups having numerosities .
The number of possible partitions is
Example The number of possible ways to assign six individuals to three teams of two people is
Factorials have numerous important applications in the analysis of probability distributions.
For example, they appear in the probability mass functions of:
the Poisson distribution;
The concept of factorial can be extended using the Gamma function
Unlike the factorial, the Gamma function is defined also when is not an integer.
It has the property thatwhen is an integer.
The Gamma function is often used in statistics, for example, in the probability density functions of:
the gamma distribution.
In turn, the Gamma function is used to define the Beta function, which is found in the densities of
the beta distribution;
the F distribution;
the T distribution.
An in-depth explanation of factorials can be found in the lecture entitled Permutations.
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Please cite as:
Taboga, Marco (2021). "Factorial", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/factorial.
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