The likelihood function is the probability density function (pdf) viewed as a function of the parameters. The maximum likelihood estimates (MLEs) are the parameter estimates that maximize the likelihood function for fixed values of x.

The maximum likelihood estimators of a and b for the gamma distribution are the solutions to the simultaneous equations

loga^−ψ(a^)=log(x¯/(∏i=1nxi)1/n)b^=x¯a^

where x¯ is the sample mean for the sample x1, x2, …, xn, and Ψ is the digamma function psi.

To fit the gamma distribution to data and find parameter estimates, use gamfit, fitdist, or mle. Unlike gamfit and mle, which return parameter estimates, fitdist returns the fitted probability distribution object GammaDistribution. The object properties a and b store the parameter estimates.

For an example, see Fit Gamma Distribution to Data.