Chi-square to standard normal distribution

[forget_f1]

Hi, I have a question

If X1,X2,...,Xn are independent random variables having chi-square distribution witn v=1 and Yn=X1+X2+...+Xn, then the limiting distribution of

(Yn/n) - 1
Z= --------------- as n->infinity is the standard normal distribution.
sqrt(2/n)

I know that Yn has chi-square distribution with v=n, but how to proceed.

 

[TenaliRaman]

Not given this much thought but ...

Now since u know Y is Chi-Square Variate with df = n
What is MGF of Y?
what is MGF of Z?
(Note : write MGF of Z in terms of MGF of Y ... i think that should be possible)
now take limit as n->oo
see if the MGF of Z is same as that of MGF of a standard normal distribution
then uniqueness theorem takes over and we are finished ...

-- AI