I am trying to solve the following problem:
Let $X_1,\dots, X_n$, where $n > 4$, be independent random variables such that $X_i ∼ N(i, i)$ for $i = 1, \dots, n$. Let $\bar{X} = {\frac{1}{n}}{\sum} X_i$ be the sample mean.
Calculate the probability that $X_n$ is not the largest observation in the sample. This probability should be expressed in terms of $\Phi (\cdot)$, the cdf of the standard normal distribution.
I am thinking we need to find $P(X_n\leq X_N )\,where\ X_N $ is largest observation. $P(X_n\leq X_N )=P(X_1\leq X_n,X_2\leq X_n,........(X_n\leq X_N)$
$ \hspace{2.7cm} =(P(X_1\leq X_n)P(X_2\leq X_n)........P(X_n\leq X_N))$
But then i lost the path. what should I do? And I am also not sure about my calculation. Any type of suggestion will be appreciated.