Statistics/Distributions/Gamma

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Contents

Gamma DistributionEdit

Gamma
Probability density function
 
Cumulative distribution function
 
Parameters
  •   shape
  •   scale
Support  
PDF  
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 FunctionEdit

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

 
 

MeanEdit

 

Now we let y=x/a which means that dy=dx/a.

 
 
 

We now use the fact that  

 

VarianceEdit

We first calculate E[X^2]

 

Now we let y=x/a which means that dy=dx/a.

 
 
 

Now we use calculate the variance

 
 

External linksEdit