Can't manipulate RV like that when it comes to distributions. Best way is to go through the moment generating function. You should expect to get another normal distribution .
1. Let Xn denote the mean of a random sample of size n from a distribution that is . Find the Limiting distribution of Xn.
2. Let Y1 denote the first order statistic of a random sample of size n from a distribution that has the PDF . Let . Find the distribution of Zn.
1.
So to find the limiting distribution, I need to take the limit of the PDF as n approaches infinity.
Kinda stuck here.
Any help would be appreciated.
No.
Post #2 told you the distribution. Your job is to understand why this is the distribution. Your job is also to review how to get the moment generating function of a sum of independent random variables.
Then you have to take the limit of the distribution as n --> +oo. The result is unsurprising (Google Dirac delta function).
This question has uncovered several apparent fundamental gaps in your knowledge and understanding. Some of these gaps have been pointed out and you must now take the necessary steps to address this.