# Finding the Variance of (1/S)

• Nov 5th 2011, 11:13 AM
jameselmore91
Finding the Variance of (1/S)
Given that we have $\displaystyle X_{1}, ... , X_{n} \sim iid G$ where $\displaystyle G$ is unknown.

Find$\displaystyle V(\overline{X} / S)$ Where $\displaystyle X$ is the sample Mean, $\displaystyle S^2$ is the sample variance and $\displaystyle V( \cdot )$ is the variance.
So...
$\displaystyle V(\overline{X} / S) = E[\overline{X}^2 / S^2] - E[\overline{X} / S]^2$
Since $\displaystyle \overline{X}$ and $\displaystyle S^2$ are independent,
$\displaystyle V(\overline{X} / S) = E[\overline{X}^2]E[1 / S^2] - E[\overline{X} / S]^2$

We were having a real hard time finding $\displaystyle E[1 / S^2]$. Also, can the $\displaystyle E[\overline{X} / S]^2$ be broken up in a similar way to the other term?

Is there another way to approach this problem?
• Nov 5th 2011, 11:16 AM
jameselmore91
Re: Finding the Variance of (1/S)
I can't get the LaTex to work. But the first error is the variance of the sample mean divided by the sample standard deviation. We've been haveing a hard time finding the E(1/S) and E(1/S^2)... Any ideas?
• Nov 5th 2011, 11:37 AM
Plato
Re: Finding the Variance of (1/S)
Quote:

Originally Posted by jameselmore91
We've been asked to find the $\displaystyle Var( \bar{X} / S)$ where $\displaystyle \bar{X}$ is the Sample Mean and S is the Sample Variance.

We've worked it all out but are having a hard time caclulating $\displaystyle Var(1/S)$ and $\displaystyle Var((1/S)^2)$.
Any ideas? Is there an easier way to approach this problem?

Use [TEX]...[/TEX] raps for LaTeX.
• Nov 5th 2011, 03:51 PM
Moo
Re: Finding the Variance of (1/S)
Without any further asumption on the rv's distributions, it's not possible to say anything...
• Nov 6th 2011, 08:56 AM
jameselmore91
Re: Finding the Variance of (1/S)
The only thing known about the random variables is that they are identically and independently distributed with population mean µ and variance "sigma squared"