In: Statistics and Probability
Let (Un, U, n>1) be asequence of random variables such that Un and U are independent, Un is N(0, 1+1/n), and U is N(0,1), for each n≥1.
Calculate p(n)=P(|Un-U|<e), for all e>0.
Please give details as much as possible
Given: and that they are independent. Hence is also normally distributed.
Let
So, .
We want:
I did all the above manipulation because . Now we can write the probabilities in terms of the standard normal distribution .
Here is the CDF of the standard normal distribution Z.