# First order statistics (min) of n non-identical but independent normal variates [duplicate]

While I have seen papers and posts about mean and variance of n i.i.d normal random variables, I am looking for the first order statistics of $n$ (specifically $11$) normal, non-identical (different mean and variances) but independent random variables. Specifically :

1. What distribution would that follow? Can it be approximated to known distributions?
2. What would be the mean and variance?

I am currently using Monte-carlo simulation for finding this, but would like a more direct value to save computational time.

• See this answer which has at least some discussion of dealing with the maximum (which is fairly easily converted to the equivalent case for the minimum). – Glen_b -Reinstate Monica Sep 6 '14 at 9:16
• Some indication of just how messy the situation is for order statistics of non-identically distributed independent variables is given at stats.stackexchange.com/questions/41438. You can save computational time by performing numerical integration--MC simulation, after all, is just a (very compute-intensive) form of integration. – whuber Sep 6 '14 at 19:02
• @AlecosPapadopoulos That's good, but if you're going to call it a duplicate, you should probably briefly outline in that answer why that answers this question as well. I think it would be helpful to do so, since the connection may not be obvious to a novice reader. Even a couple of sentences explaining the connection might suffice. – Glen_b -Reinstate Monica Sep 7 '14 at 1:49
• @Glen_b You're right. I added some text in the beginning of my answer there, and a reminder when the density is derived. – Alecos Papadopoulos Sep 7 '14 at 2:14
• Your title doesn't seem to match the questions in the body of the post. Do you want to know about the minimum, or about the distribution more generally (type, mean & variance, etc)? Are you asking about the distribution of a variable that is a function (eg, the sum) of $n$ variables or the distribution of the mixture of $n$ variables? – gung - Reinstate Monica Sep 7 '14 at 15:41

Let's denote the PDF and CDF of the standard normal distribution as $\phi(x)$ and $\Phi(x)$, and suppose that there are $n$ normally distributed random variables with means and standard deviations of $\{\mu_{i}\}$ and $\{\sigma_{i}\}$. Then, the PDF of first-order statistics should be $$p(x) = \sum_{i=1}^{n} \frac{1}{\sigma_{i}}\phi(\tfrac{x-\mu_{i}}{\sigma_{i}})\prod_{\substack{j=1\\j\neq i}}^{n}\Big[1- \Phi(\tfrac{x-\mu_{j}}{\sigma_{j}})\Big].$$ I can't see any obvious way to further simplify this expression.