We present a full Bayesian inference for the parameter and the associated reliability function of the Exponentiated Gamma distribution. This distribution has the advantage of being uni-parametric and its hazard function takes on increasing and bathtub shapes. We show that dealing with Jeffreys, uniform and gamma priors for the parameter then the corresponding posterior distributions result in a Gamma. Besides, the posterior distributions of cumulative function for a fixed lifetime are Negative Log-Gamma (NLG) distributions under transformation of parameter and also when Jeffreys, uniform and NGL priors are assigned to the cumulative function. This way, we show that the families of gamma and NLG distributions provide conjugate prior distribution for the parameters of interest. We also construct credible and HPD intervals in a simple closed-form. We further derive the predictive distribution and interval of future observation assuming gamma posterior distribution for the parameter.