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

 

[tacman]

To understand why the limiting distribution of Z is the standard normal distribution, we need to understand the central limit theorem. The central limit theorem states that as the sample size n increases, the sampling distribution of the mean of n independent and identically distributed random variables will approach a normal distribution, regardless of the underlying distribution of the individual variables.

In this case, we have n independent random variables with chi-square distribution and we are interested in the distribution of the sample mean, Yn/n. By the central limit theorem, as n approaches infinity, the distribution of Yn/n will approach a normal distribution with mean and standard deviation given by the mean and standard deviation of the underlying chi-square distribution.

Using the properties of chi-square distribution, we can calculate that the mean of Yn/n is 1 and the standard deviation is sqrt(2/n). Therefore, as n approaches infinity, the distribution of Z = (Yn/n - 1)/sqrt(2/n) will approach a standard normal distribution with mean 0 and standard deviation 1. This is because we are subtracting 1 from the mean and dividing by the standard deviation of Yn/n, which will normalize the distribution to a standard normal distribution.

In summary, the limiting distribution of Z is the standard normal distribution because of the central limit theorem and the properties of chi-square distribution.