Finding the Variance of (1/S)

**jameselmore91** · November 5th 2011, 11:13 AM

Given that we have $X_{1}, ... , X_{n} \sim iid G$ where $G$ is unknown.

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

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

Is there another way to approach this problem?

**jameselmore91** · November 5th 2011, 11:16 AM

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?

**Plato** · November 5th 2011, 11:37 AM

Originally Posted by jameselmore91

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

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

Use [TEX]...[/TEX] raps for LaTeX.

**Moo** · November 5th 2011, 03:51 PM

Without any further asumption on the rv's distributions, it's not possible to say anything...

**jameselmore91** · November 6th 2011, 08:56 AM

The only thing known about the random variables is that they are identically and independently distributed with population mean µ and variance "sigma squared"

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