# $\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

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.

• Your question is undefined. If, by standard deviation, you intend the square root of the sample variance, you need to define what estimator of sample variance you intend to use. There are two versions in common usage. – wolfies Nov 26 '14 at 17:45
• what is the other version of usage ? – moksha Nov 26 '14 at 18:31
• Other is brother to another mother. – wolfies Nov 26 '14 at 18:43
• can we say that the equation in the question shown as $\mathsf E[T^k]$ is the kth moment of the standard deviation and the condition for it to exist is that k<3. Since the first moment will be the expectation of the standard deviation, the second moment will be the expectation of the variance. Third moment of $\mathsf \sigma$ doesnt exist. – moksha Nov 27 '14 at 16:27
• How do we find out what distribution it follows. – moksha Nov 27 '14 at 16:55