Statistics/Distributions/Gamma
Contents
Gamma Distribution[edit]
Probability density function 

Cumulative distribution function 

Parameters  

Support  
CDF  
Mean  (see digamma function) 
Median  No simple closed form 
Mode  
Variance  (see trigamma function ) 
Skewness  
Ex. kurtosis  
Entropy 
The Gamma distribution is very important for technical reasons, since it is the parent of the exponential distribution and can explain many other distributions.
The probability distribution function is:
Where is the Gamma function. The cumulative distribution function cannot be found unless p=1, in which case the Gamma distribution becomes the exponential distribution. The Gamma distribution of the stochastic variable X is denoted as .
Alternatively, the gamma distribution can be parameterized in terms of a shape parameter and an inverse scale parameter , called a rate parameter:
where the constant can be calculated setting the integral of the density function as 1:
following:
and, with change of variable :
following:
Probability Density Function[edit]
We first check that the total integral of the probability density function is 1.
Now we let y=x/a which means that dy=dx/a
Mean[edit]
Now we let y=x/a which means that dy=dx/a.
We now use the fact that
Variance[edit]
We first calculate E[X^2]
Now we let y=x/a which means that dy=dx/a.
Now we use calculate the variance