# Calculating Variances of Functions of Sample Mean

Scootertaj
1. Essentially what I'm trying to do is find the asymptotic distributions for
a)
Y2
b) 1/Y and
c) eY where
Y = sample mean of a random iid sample of size n.
E(X) = u; V(X) = σ2

## Homework Equations

a) $$Y^2=Y*Y$$ which converges in probability to $$u^2$$,

$$V(Y*Y)=\sigma^4 + 2\sigma^2u^2$$

So, $$\sqrt{n}*(Y^2 - u^2)$$ converges in probability to $$N(0,\sigma^4 + 2\sigma^2u^2)$$

So, $$Y^2\sim N(u^2,\frac{\sigma^4 + 2\sigma^2u^2}{n})$$

Is that right?

b) $$\frac{1}{Y}$$ converges in probability to $$\frac{1}{E(X)} = \frac{1}{u}$$
$$V(\frac{1}{x}) = \frac{1}{σ^2}$$ ??

Thus,
$$\sqrt{n}(\frac{1}{Y} - \frac{1}{u})$$ converges in distribution to $$N(0,V(\frac{1}{x})*\frac{1}{n})$$
What is V(1/X) ?

Am I on the right track?

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