A generalization of the log-likelihood function and weighted average in Gauss' law of error

One of the promising approaches to how to derive a non-Gaussian distribution is generalizing the log-likelihood function in Gauss' law of error. In this contribution, it is shown that a generalization of the log-likelihood function in Gauss' law of error is equivalent to a generalization of the average. The proof is given for the case of the two-parameter generalized likelihood function, which unifies some known one-parameter generalizations.