1. ## Central Limit Theorem

Let X1,X2,… be a sequence of independent N(0, 8) distributed random variables. For n=1,2,… let Yn be the random variable defined by

Yn=X^2(subscript)1+⋯+X^2(subscript)n.
Use the central limit theorem rule of thumb to approximate P(Y25>216). You can use that E[X^4(subscript)i]=192.

I'm really struggling with how to do this question, I tried to use the formula for Var(x)=e[x^2] - [E[x]]^2
and tried standardising using Zn = (Xn(bar) - E[x])/ sigma/root(n) but it just doesn't seem to be working. Any help and explanations would be greatly appreciated.

2. ## Re: Central Limit Theorem

Originally Posted by cs1632
Let X1,X2,… be a sequence of independent N(0, 8) distributed random variables. For n=1,2,… let Yn be the random variable defined by

Yn=X^2(subscript)1+⋯+X^2(subscript)n.
Use the central limit theorem rule of thumb to approximate P(Y25>216). You can use that E[X^4(subscript)i]=192.

I'm really struggling with how to do this question, I tried to use the formula for Var(x)=e[x^2] - [E[x]]^2
and tried standardising using Zn = (Xn(bar) - E[x])/ sigma/root(n) but it just doesn't seem to be working. Any help and explanations would be greatly appreciated.
ok, you've got a sum of the squares of iid normal rvs each $\sim N(0,\sigma^2)$.

The first step is to determine the mean and variance of each of these.

We find that

$E[Y_1] = \sigma^2$

$Var[Y_1] = 2\sigma^4$

so by the central limit theorem

$\sqrt{25}\left(\left(\dfrac{1}{25} \displaystyle{\sum_{k=1}^{25}}~X^2_k\right)-\sigma^2\right) \Rightarrow N(\sigma^2, 2\sigma^4)$

you should be able to complete the algebraic massage needed to answer your problem.

3. ## Re: Central Limit Theorem

Originally Posted by romsek
ok, you've got a sum of the squares of iid normal rvs each $\sim N(0,\sigma^2)$.

The first step is to determine the mean and variance of each of these.

We find that

$E[Y_1] = \sigma^2$

$Var[Y_1] = 2\sigma^4$

so by the central limit theorem

$\sqrt{25}\left(\left(\dfrac{1}{25} \displaystyle{\sum_{k=1}^{25}}~X^2_k\right)-\sigma^2\right) \Rightarrow N(\sigma^2, 2\sigma^4)$

you should be able to complete the algebraic massage needed to answer your problem.
my mistake

$\sqrt{25}\left(\left(\dfrac{1}{25} \displaystyle{\sum_{k=1}^{25}}~X^2_k\right)-\sigma^2\right) \Rightarrow N(0, 2\sigma^4)$