Would absolutely love if someone could help me with this question, in a step by step way to help those who are uninitiated to Statistics and Mathematics.
So, I am trying to "prove/justify" through MGFs how as n(the sample size) increases and goes to infinity, a standardized binomial distribution converges to the Standard Normal Distribution. So in the beginning we have $X_n $~$ Bin(n,p)$ then we standardize this $R.V.$ by subtracting the mean $E(X_n)=np$ and dividing by the $SD$ which is $np(1-p)$. After doing this, we get a new $R.V.$ with $mean=1$ and $variance=SD=1$. It is the distribution of this standardized quantity that converges to a fixed distribution, correct? Namely the Standard Normal Distribution. Let's call the standardized version of $X_n$, $Z_n$.
I think what we need to do is to find the MGF of $Z_n$ and show that this MGF becomes the Standard Normal Distribution's MGF as the sample size $(n)$ heads toward infinity. Now $Z_n$ is a function of $X_n$ which we know the MGF of which is $(1-p+pe^t)^n$.
Now I know that $Z_n$ is a linear transform and that we should be able to use this to simply the MGF but I don't understand how the course notes do it, mostly because they skip a lot of steps and it is difficult to follow. There is also a Taylor Series Expansion for the exponential that I am confused about although I do understand the concept of the series expansion.
Help would be greatly appreciated! I would love assistant in exactly how we can prove this via MGF.
Thanks so much in advance :)