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### What is the mean and variance of the median of a set of i.i.d normal random variables?

Let $X_1$, ..., $X_n$ be identically independently distributed random variables with $N(\mu, \sigma^2)$ It is easy to show that sample mean $\bar{X} = \frac{1}{n}\sum^n_{i = 0}{X_i}$ is a random ...
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### Assuming two Gaussian distributions of equal mean and variance, then how different can we expect the top X members of each group to be?

Here's the thread I got the idea from: http://www.quora.com/Do-men-have-a-wider-variance-of-intelligence-than-women/answer/Ed-Yong Basically, this is a model that might be able to explain why there ...
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### Distribution of maximum of normally distributed random variables

I'm trying to find the closed-form CDF and PDF of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. My thought process so far:  \begin{align*} F_Y(y) &= \mathbb{P}(\max(...
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### Explanation for this event on a high-dimensional dataset

Suppose we sample a set $S$ of $n$ points from a $d$-dimensional spherical (unit variance) Gaussian with $d \approx 100$. It is known that any point of the sample would be roughly at $\sqrt{d}$ ...
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### Variance of Normal Order Statistics

Suppose we have $X_1, \cdots, X_n \overset{\textrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$ with $n > 50$, and let $X_{(1)}, \cdots, X_{(n)}$ be the associated order statistics. Are there any references ...
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### Expected value of maximum of samples from normal distribution

Lets say I have a normal distribution $N(\mu, \sigma^2)$ from which I have drawn $n$ i.i.d. samples $x_1, \dots, x_n$. Now, lets define a random variable $Y = max(x_1, \dots, x_n)$. When $n=1$, the ...
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### Approximate Order Statistics for lognormal variables

Are there any known formulas that approximate the expected value of the maximum of $N$ i.i.d. lognormal random variables? I am looking for something similar to: Approximate order statistics for ...
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### Approximate variance for 99.5th percentile for normal distribution

My question is similar to this one: Approximate order statistics for normal random variables I am looking to find a formula for the variability of an arbitrary percentile of a normal distribution. ...
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### expected lowest value of 10 normally distributed values

Consider 10 values that follow a standard normal distribution. What would you expect to be the lowest value? I tried to simulate this problem in R. I basically just simulated 100000 standard normal ...
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### In R, what does qqnorm actually do?

qqnorm(x) plot(qnorm(seq(80)/81),sort(x)) Finding that the plots produced by the commands above are slightly different from each other, I tried this: ...
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### Determine the limiting distribution of Standard Normal order statistics

Let $X_1,...,X_n$ be an i.i.d sample from the standard normal distribution. Is there any general formula for the first order statistic of this sample? For example, I have seen a formula by Blom (...