# Gamma distribution and hyperparameters

The formula for mean and variance of a gamma distribution is given by a/b and a/b^2 (hyperparameters) respectively.Are they estimates of the posterior gamma distribution?

Can prior, likelihood and posterior all 3 have gamma distribution, and can their means and variances computed using the formula above?

If your likelihood function is a Gamma with known shape, $$a$$, and the parameter of interest is the rate, $$b$$, then the Gamma distribution is a conjugate prior - see https://en.wikipedia.org/wiki/Conjugate_prior#When_likelihood_function_is_a_continuous_distribution
In Bayesian analysis the parameters are uncertain, the data is not - i.e. we are interested in $$P[parameters | data]$$.
As you correctly say, any Gamma distribution is uniquely specified my its mean and variance: you can calculate these from $$a$$,$$b$$ using the formula you give, or you can use the mean and variance to estimate $$a$$,$$b$$ by reversing the equations you quote.