1. ## Central Limit Theorem

See Question 3 on the attached document....

It is a past exam paper but the solutions are not provided so i cant see where i am going wrong....

I derived the PGF in part (i) then proceeded to get E(x) as 11/5 and Var(x) = 39/25

The problem is in part (iii) in the central limit theorem. I constructed the sample mean as (11/5)*50 to give 110 and the variance as $\displaystyle {\frac {\sigma}{\sqrt {n}}}$ which i got as such a small value (sqrt39/25)/(sqrt50) that it made my Z values so large that phi of these values were like 56 and -320 which would obviously be approximately 1-0, meaning almost the whole of the set is in this interval.... anyway it didnt seem right! have i gone wrong somewhere?

2. Have i gone wrong on part (iii)?

3. I'm not sure what you're asking.

I get $\displaystyle E(X)=2.2$ and $\displaystyle E(X^2)=6.4$

So $\displaystyle V(X)=1.56$

Thus $\displaystyle Y\approx N((50)(2.2), (50)(1.56))$

Since the X's are integer based you may want to put in a correction factor, but that won't