# Posterior predictive distribution example

Assume there's some normally distributed population ($$X$$) whose parameters ($$\mu$$, $$\sigma$$) are not known. A sample ($$x_1$$) of size $$n$$ is drawn from $$X$$, and statistics are calculated: $$\bar{x}_1$$ and $$s_1$$. Another sample, $$x_2$$ of size $$m$$, has yet to be drawn but will be.

I'd like use this information ($$n$$, $$\bar{x}_1$$, $$s_1$$, and $$m$$) to formulate closed-form expectations of $$\bar{x}_2$$ and $$s_2$$, and, after some research, it seems like posterior predictive distributions are the right tool here, specifically when paired with the Normal-gamma conjugate prior.

Could someone confirm this intuition and/or provide an example of using something like this?

• What do you mean by a sample of length m? Can you be more concrete? – activatedgeek Jun 24 at 21:27
• @activatedgeek I should have said size instead of length. Does that clear it up? – cjken Jun 24 at 22:18