An often-cited example of the gamma distribution is that for integer shape parameter k, and scale parameter lambda, the gamma distribution can be conceptualized as the sum of k independent (identically) exponentially-distributed random variables with termination rate lambda. My question is this: what does this shape parameter mean when it is not an integer? I am a graduate student in psychology trying to model termination rates for behavioural activities in a hidden markov model, and if the "best fitting" gamma distribution has non-integer shape parameter, I would like to know how to decompose the gamma into a sum of exponential processes which, in the non-integer case, would presumably have different termination rates. Any help would be greatly appreciated. Thanks!

An often-cited example of the gamma distribution is that for integer shape parameter k, and scale parameter lambda, the gamma distribution can be conceptualized as the sum of k independent (identically) exponentially-distributed random variables with termination rate lambda. My question is this: what does this shape parameter mean when it is not an integer? I am a graduate student in psychology trying to model termination rates for behavioural activities in a hidden markov model, and if the "best fitting" gamma distribution has non-integer shape parameter, I would like to know how to decompose the gamma into a sum of exponential processes which, in the non-integer case, would presumably have different termination rates. Any help would be greatly appreciated.