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If $X_i, i=1,...,$ are independent, identically distributed $\operatorname{N}(0,1)$ random variables, and $Y_i = X_i^2$ are independent $\operatorname{Gamma}(\frac{1}{2},\frac{1}{2})$ RVs, use the Central Limit Theorem to calculate the approximate values of the probabilities:

$\sum_{1}^{50} {Y_i \gt 40}$ and $\sum_{1}^{50} {Y_i \lt 60}$

I'm not really sure how to go about this.

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    $\begingroup$ Hi Mathlete, this appears to be standard bookwork and looks like it's probably related to some course. Would you mind adding the self-study tag please? [You might like to read through the faq item relating to homework (a synonym for self-study) and the self study tag wiki for a description of how those questions are dealt with. In particular, there are some expectations of both askers and answerers that differ slightly from the other kinds of questions.] $\endgroup$ – Glen_b Apr 23 '13 at 0:06
  • $\begingroup$ Because $E(Y_i)=E(X_i^2)=E((X_i-0)^2)$ is the variance of $X_i$, clearly the mean of $Y_i$ is $1$. Because $E((Y_i-1)^2)=E(Y_i^2)-2E(Y_i)+E(1)=E(X_i^4)-2+1=2$, the variance of $Y_i$ is $2$. That's all you need to solve this problem. $\endgroup$ – whuber Apr 25 '13 at 13:23
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It's been a couple of semesters since the delta method, so bear with me here.

What does the CLT tell us?

$\frac{\sqrt{n}(\bar{x}-\mu)}{\sigma} \rightarrow N(0,1)$

You can use the delta method to determine the asymptotic distribution of $X^2$.

$\sqrt{n}(g(X_n) - g(\mu)) \rightarrow N(0,\sigma^2(g'(\mu))^2)$

So you need to set your $g(X)$, in this case $X^2$. $g(\mu) = \mu^2$. you know the distribution of $X^2$ so you know what the $\sigma^2$ is. From there you should be able to turn this into a N(0,1) and use z-tables to figure out what the probability of $\Sigma Y_i > 40$ and $\Sigma Y_i < 60$. Hope this helps.

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  • $\begingroup$ So I would get $\sqrt{n}$$(X_n^2)$ -> $N(0,(g'(0))^2)$ How would I follow on from there? $\endgroup$ – Mathlete Apr 22 '13 at 23:02
  • $\begingroup$ Considering I get $g'(\mu)=0$ have I done something wrong? $\endgroup$ – Mathlete Apr 22 '13 at 23:07
  • $\begingroup$ no. $g(\mu) = \mu^2$ so $g'(\mu) = 2\mu$, $g'(\mu)^2$ = $4\mu^2$ $\endgroup$ – Eric Peterson Apr 22 '13 at 23:08
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    $\begingroup$ bmbolstad.com/teaching/Stat215b/Lab4/delta_method.pdf Here are a few examples on how to carry out the delta method and get a normal distribution asymptotically. Hope this helps. Once you've seen a few carried out, maybe you can solve this one. Also, look at the wiki page for delta method: en.wikipedia.org/wiki/Delta_method $\endgroup$ – Eric Peterson Apr 22 '13 at 23:24
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    $\begingroup$ @ClarkW.Griswold my guess is they've probably just done the CLT, and haven't even mentioned the Delta method. The CLT is sufficient for this question. $\endgroup$ – Glen_b Apr 23 '13 at 0:01

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