Approximate distribution of normal squared I am studying for a test, one section of which will cover the delta method. This problem came from that section:
Let $X\sim N(\mu,n^{-1})$. Find an approximate distribution of $X^2$. (It also asks for exact distribution, but I can do that). 
Can you help me get started? I can't see how I can apply delta method or central limit theorem etc. to find some approximation. 
I know that $\sqrt{n}(X-\mu) \sim N(0,1)$, but I don't see how that can help either. 
 A: Let's double check what the delta method is:

Roughly, if there is a sequence of random variables Xn satisfying
  $$
    {\sqrt{n}[X_n-\theta]\,\xrightarrow{D}\,\mathcal{N}(0,\sigma^2)},
$$
  where $\theta$ and $\sigma^2$ are finite valued constants and $\xrightarrow{D}$ denotes convergence in distribution, then
  $$
    {\sqrt{n}[g(X_n)-g(\theta)]\,\xrightarrow{D}\,\mathcal{N}(0,\sigma^2[g'(\theta)]^2)}
$$
  for any function $g$ satisfying the property that $g′(\theta)$ exists and is non-zero valued.

Okay, so showing you have the first thing already is easy right?
So let $g(x) = x^2$ and you're set, aren't you? Just apply the theorem.
Then at the end do the relevant linear transformation (affecting variance and mean) so that you're just talking about what $g(X)$ will approximately be distributed as.
So in short:
i) State the theorem.
ii) Explain/show how the first part is okay
iii) state $g$
iv) apply the theorem
v) infer approximate distribution for $g(X)$
Which is pretty much what you do any time you want to use it.
