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I have two set of gamma distribution .

standard deviation is related to its population mean

Given unequal standard deviation, how do I make sure the population mean is equal (null hypothesis is TRUE) ?

Is the answer is we have to minus the population mean (alpha*beta) from the every sample we pick from simulation.? [ since i am care about mean and variance, then how can i do to maintain the populatio mean

Scale parameters -- How do they work, why are they sometimes dropped?

If you care at all about variances being equal, you care about the scaling parameter. . In the general case, if you care about means, you care about scaling. [credited from post above]

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  • $\begingroup$ Please be explicit about whether your $\beta$ is intended to be a scale or rate parameter (or indeed, something else). Further, your question seems to largely be answering itself (you state things that largely seem to answer the questions you ask) $\endgroup$ – Glen_b -Reinstate Monica Apr 11 '16 at 17:26
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    $\begingroup$ This reads like a reworded version of a routine textbook-style exercise. Is this work for some class? $\endgroup$ – Glen_b -Reinstate Monica Apr 11 '16 at 17:31
  • $\begingroup$ erm...Since i am given five skewness, I did tried to use formula of skewness to find out the shape parameter and then try to use standard deviation to find scale parameter, however, the answer is not the same in two distribution... $\endgroup$ – j.l Apr 12 '16 at 2:32
  • $\begingroup$ i am sorry but it is not a work, just need an further understanding on gamma distribution and define them by myself>< Because i couldnt find online and books . $\endgroup$ – j.l Apr 12 '16 at 2:34
  • $\begingroup$ I don't quite understand the situation you're describing in your second-last comment. Can you explain the circumstances in more detail (perhaps in your question)? (if you want to notify me put @Glen_b in a comment to tell me you did so -- I won't otherwise know when your question is edited.) $\endgroup$ – Glen_b -Reinstate Monica Apr 12 '16 at 2:48
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  1. As you state in your question, the moment skewness of a gamma distribution is $\frac{2}{\sqrt{\alpha}}$, where $\alpha$ is the shape parameter.

    This means that if you know the two population shape parameters, and they're the same then the two distributions have the same moment-skewness.

  2. The symbol $\beta$ is sometimes used for the scale parameter of the gamma (let's call that $\tau$), and sometimes for the rate parameter (let's call that $\lambda$). Since it's not presently clear which you seek, I'll discuss both.

    The variance of a gamma distribution is $\alpha \tau^2=\alpha /\lambda^2$ from which you can find the standard deviation.

    [Both these facts are on the Wikipedia page for the distribution.]

    So the standard deviation is related to the skewness parameter (since the standard deviation is a function of $\alpha$, if you change the skewness while holding the other parameter constant you will change the standard deviation) but you can change the standard deviation without changing the skewness, by changing the scale parameter, which doesn't affect the skewness.

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