Probability distributions conditioned by the available information: Gamma distribution and moments

Given a gamma probability distribution g as the observed distribution, and the information available on moments of the random variable, the probability distribution ƒ is derived such that the χ2-distance between ƒ and g is minimum. The explicit expressions for the density function, minimum χ2-measure and moments are given. Since in actual applications, the available information is in the form of observed frequency distribution and sometimes on mean, geometric mean and/or variance of random variable, these cases are considered in detail, followed by a numerical illustration.