0
votes
1answer
21 views

Proving some properties about the expected first order statistic (maximum) with respect to sample size.

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as ...
0
votes
1answer
29 views

How to calculate $\int\limits_{-\infty}^{+\infty} \frac{n-1}{\alpha}\Phi(\frac{x}{\alpha})^{n-2}\phi(\frac{x}{\alpha})^2dx$?

I was working on a research project that involves taking the integral of $$\frac{n-1}{\alpha}\int\limits_{-\infty}^{+\infty} ...
0
votes
0answers
24 views

Normal Distribution while finding sigma

I was reading some things about normal distribution and saw this problem in a text a couple days ago. I know it might be a little advanced for me at the moment, but I was wanted to know if someone can ...
4
votes
2answers
2k views

Distribution of the maximum of a multivariate normal random variable

Suppose there is a vector of jointly normally distributed random variables $X \sim \mathcal{N}(\mu_X, \Sigma_X)$. What is the distribution of the maximum among them? In other words, I am interested in ...
22
votes
2answers
7k views

Expectation of the maximum of gaussian random variables

Is there an exact or good approximate expression for the expectation, variance or other moments of the maximum of $n$ independent, identically distributed gaussian random variables where $n$ is large? ...