What is the pdf of sample mean and variance, given the population mean and variance? Suppose I have drawn n samples from a population of known mean and variance ( for example, a normal distribution with mean zero and variance 1.0 ).
I then calculate the mean and standard deviation of the sample.
How do I calculate the pdf of these sample values, given that I know the population values?
 A: The distributions of the sample mean and variance of a normal distribution are well-known (normal for the mean, Chi square for the variance).  As whuber says, you can't find the pdfs of the sample mean $\overline{x}$ and, especially, the variance $s^2$ except in special situations. Given only the population mean $\mu$ and variance $\sigma^2$ and nothing else, all you can find exactly are sample mean and variance of  $\overline{x}$ and the mean of $s^2$ (but not $s$):
Let the sample size be $n$. Then every introductory text on statistical theory  demonstrates that:
$$ 
E(\overline{x})= \mu
$$
$$
Var(\overline{x}) = \frac{\sigma^2}{n}
$$
and
$$
E(s^2) = \sigma^2
$$
If you know, in addition to $\mu$ and $\sigma^2$, the population fourth central moment $ \mu_4 = E[(X =\mu)^4]$, you can also compute the exact variance of $s^2$  
$$Var(s^2) = \frac{(n-1)^2}{n^3}\left(\mu_4 - \frac{n-3}{n-1}\sigma^2\right)
$$
Reference: 
CR Rao (1965) Linear Statistical inference and its applications, Wiley, New York, p.368.
