Gamma distribution parameters

[panbalag]

Hi,

I'm trying to understand the theory behind the probability distributions specifically what do the shape and scale parameters for gamma distribution mean. I started off with the following material

http://mathworld.wolfram.com/GammaDistribution.html

It is given that alpha parameter is similar to h (which is a count of poisson events). If this is the case, then why should the value of alpha be between 0 and 1 to indicate a decreasing failure rate? Can't the number of events be any positive value?

Can someone give me pointers to the interpretation of these parameters?

Regards
Prasanth

 

[royhaas]

The time between Poisson events has an exponential distribution, which is a Gamma distribution with \(\displaystyle \alpha = 1\). The exponential distribution has a constant failure rate (lookup the definition of failure rate). By definition, a Gamma distribution with \(\displaystyle \alpha < 1\) has an increasing failure rate, while \(\displaystyle \alpha > 1\) gives a decreasing failure rate.

 

[panbalag]

What does alpha mean in the context of decreasing or increasing failure rate ?