Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the existence of this moment.
Basically we are being asked to find $\mathsf E[T^k]$. So should I do something by finding the distribution of the standard deviation or can we infer something from the behaviour of the random variable of interest.
Can we write it as,
$\mathsf E[\sigma^k] = \int \sigma^k pdf d\sigma$
if its true how do we find out the distribution of the standard deviation. Will the standard deviation also follow a normal distribution.