Thank you very much CrazyStat. I was convinced that "x-bar and s are independent for data coming from a Normal distribution". So the conclusion is for my exampl

Thank you very much CrazyStat. I was convinced that "x-bar and s are independent for data coming from a Normal distribution". So the conclusion is for my example, p(x|mu) is normal distribution, p(x-bar|mu), after derivation, is also normal distribution. If p(x|mu) comes from some other distributions, p(x-bar|mu), however, might not be normal. Thanks for providing the proof, and I will only be convinced by the proof :) Will update the post shortly. Thanks for the great conversation and suggestions.