# Is it correct to say the gamma distribution is completely determined by its first two moments?

I have heard this said by one of my professors, but he is not available for me to ask anymore. But if this is the case then why are there others defined on the wikipedia page, and why do authors refer to higher order moments in papers??

Many thanks.

• So the Gamma distribution is completely defined by its first two moments? – user124123 Apr 30 '13 at 21:32
• The gamma distribution is defined by its first two moments in the sense that specifying the first two moments (and the fact that it's gamma) is sufficient to determine the two parameters, since the first two moments depend on the two parameters in such a way as to make it possible to solve for them. That doesn't imply that the higher order moments don't exist, any more than it does for the normal distribution, so why would authors not refer to the higher order moments when they're relevant, just as they do for other distributions which are determined by their lower order moments? – Glen_b May 1 '13 at 0:55

For $X \sim \Gamma(\alpha,\beta)$ you have $E(X)=\frac{\alpha}{\beta}$ and $E(X^2)=\frac{\alpha+\alpha^2}{\beta^2}$.
You can solve these two equations in two unknowns, and recover $\alpha$ and $\beta$.