What is the connection between X_n and Y_n?

[Ted123]

Homework Statement

[PLAIN]http://img263.imageshack.us/img263/8679/statsji.jpg

The Attempt at a Solution

I've done part (a) and I know what the CLT says but how does part (a) link with part (b) as if $X_n \sim Bern(p)$ then $\displaystyle \sum^n_{i=1} X_i \sim Bin(n,p)$ so $X_n = \displaystyle \sum^n_{i=1} Y_i$ where $Y_1 , \cdots , Y_n \sim Bin(n,p)$ are i.i.d

BUT $X_n \sim Bin(n,p)$ so where do I go from here?

 

[vela]

How did you come up with

$$X_n = \sum^n_{i=1} Y_i$$

?

 

[Ted123]

How did you come up with

$$X_n = \sum^n_{i=1} Y_i$$

?

Actually it should be $Y_n = \sum^n_{i=1} X_i$ where $X_1, ..., X_n \sim Bern(p)$ (iid)

so $\frac{Y_n - \mathbb{E}[Y_n]}{\sqrt{Var(Y_n)}} = \frac{Y_n - \sum^n_{i=1} \mathbb{E}[Y_i]}{\sqrt{\sum^n_{i=1} Var(Y_i)}} = \frac{Y_n - n\mathbb{E}[Y_1]}{\sqrt{nVar(Y_1)}} = \frac{Y_n - np}{\sqrt{np(1-p)}} \to Y ;\; Y\sim N(0,1)$ by CLT