Picking values from a normal distribution?

Question

So I have the following formulas:

$\frac {\mu-\bar{x}}{\frac{\sigma}{\sqrt{n}}} \sim N(0,1)$

and

$\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{\sigma^2} \sim\chi^2_{n-1}$

and given following sample:

Sample 1: 120 , 107 , 110 , 116 , 114 , 111 , 113 , 117 , 114 , 112

Assuming I can generate values from a N(0,1), how can I sample posterior values for $\mu$ ?

EDIT: I forgot to include:

$\mu|\sigma^2 \sim N(\beta, \frac{\sigma^2}{n_o})$

$\mu\,|\,x_1,x_2,....x_n\,,\,\sigma^2 \sim N(\frac {n\bar{x} + n_o\beta}{ n + n_o} \, , \frac {\sigma^2}{n + n_o})$

Answer 1

First up, your formulas don't tell us much except that

$$ x \sim N(\mu,\sigma^2). $$

To sample from the posterior distribution $f(\mu|\bar{x})$ we'd need to know $\mu$'s prior $f(\mu)$. These are related with Bayes' law.

$$ f(\mu|\bar{x}) = \frac{f(\bar{x}|\mu) f(\mu)}{f(\bar{x})} $$