The Gamma distribution can be thought of as a generalization of the Chi-square distribution. If a random variable has a Chi-square distribution with degrees of freedom and is a strictly positive constant, then the random variable defined as: has a Gamma distribution with parameters and .
Gamma random variables are characterized as follows:
Definition Let be an absolutely continuous random variable. Let its support be the set of positive real numbers:Let . We say that has a Gamma distribution with parameters and if its probability density function is:where is a constant:and is the Gamma function.
A random variable having a Gamma distribution is also called a Gamma random variable.
To better understand the Gamma distribution, you can have a look at its density plots.
The expected value of a Gamma random variable is:
It can be derived as follows:
The variance of a Gamma random variable is:
It can be derived thanks to the usual variance formula ():
The moment generating function of a Gamma random variable is defined for any :
Using the definition of moment generating function:where the integral equals because it is the integral of the probability density function of a Gamma random variable with parameters and . Thus:Of course, the above integrals converge only if , i.e. only if . Therefore, the moment generating function of a Gamma random variable exists for all .
The characteristic function of a Gamma random variable is:
Using the definition of characteristic function:
The distribution function of a Gamma random variable is:where the functionis called lower incomplete Gamma function and is usually evaluated using specialized computer algorithms.
This is proved as follows:
In the following subsections you can find more details about the Gamma distribution.
If a variable has the Gamma distribution with parameters and , then:where has a Chi-square distribution with degrees of freedom.
This can be easily proved using the formula for the density of a function of an absolutely continuous variable ( is a strictly increasing function of , since is strictly positive):The density function of a Chi-square random variable with degrees of freedom is:where Therefore,which is the density of a Gamma distribution with parameters and .
Thus, the Chi-square distribution is a special case of the Gamma distribution, because, when , we have:
In other words, a Gamma distribution with parameters and is just a Chi square distribution with degrees of freedom.
Multiplying a Gamma random variable by a strictly positive constant one obtains another Gamma random variable. If is a Gamma random variable with parameters and , then the random variable defined as:has a Gamma distribution with parameters and .
This can be easily seen using the result from the previous subsection:where has a Chi-square distribution with degrees of freedom. Therefore:In other words, is equal to a Chi-square random variable with degrees of freedom, divided by and multiplied by . Therefore, it has a Gamma distribution with parameters and .
In the lecture entitled Chi-square distribution we have explained that a Chi-square random variable with degrees of freedom ( integer) can be written as a sum of squares of independent normal random variables , ..., having mean and variance :
In the previous subsections we have seen that a variable having a Gamma distribution with parameters and can be written as:where has a Chi-square distribution with degrees of freedom.
Putting these two things together, we obtain:where we have defined:But the variables are normal random variables with mean and variance . Therefore, a Gamma random variable with parameters and can be seen as a sum of squares of independent normal random variables having mean and variance .
Below you can find some exercises with explained solutions:
Exercise set 1 (recognize Gamma distributions and derive their parameters).
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