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### How exactly did statisticians agree to using (n-1) as the unbiased estimator for population variance without simulation?

The formula for computing variance has $(n-1)$ in the denominator: $s^2 = \frac{\sum_{i=1}^N (x_i - \bar{x})^2}{n-1}$ I've always wondered why. However, reading and watching a few good videos about "...
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### Why does one have to use REML (instead of ML) for choosing among nested var-covar models?

Various descriptions on model selection on random effects of Linear Mixed Models instruct to use REML. I know difference between REML and ML at some level, but I don't understand why REML should be ...
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### Why does Restricted maximum likelihood yield a better (unbiased) estimate of the variance?

I'm reading Doug Bates' theory paper on R's lme4 package to better understand the nitty-gritty of mixed models, and came across an intriguing result that I'd like to understand better, about using ...
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### When estimating variance, why do unbiased estimators divide by n-1 yet maximum likelihood estimates divide by n?

I am totally confused: On the one hand you can read all kinds of explanations why you have to divide by n-1 to get an unbiased estimator for the (unknown) population variance (degrees of freedom, not ...
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### Alternate (?) definition of sample variance [duplicate]

The variance of a sample can be defined as $$s^2 = \frac{1}{2}\frac{1}{n(n-1)}\sum_{i}\sum_{j\ne i}\left(x_i - x_j\right)^2$$ Apart from the factor of $1/2$, this can be paraphrased verbally as ...
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### What value of $\alpha$ makes $\sum_{i=0}^n (x_i-\alpha)^2$ minimal?

I know that $\bar{x}$ makes absolute result of $\sum_{i=0}^n (x_i-\alpha)$ minimum. In fact it makes it zero. But how to find what value of $\alpha$ makes $\sum_{i=0}^n (x_i-\alpha)^2$ minimal? What ...
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### Definition of a sample: can it include the same object twice?

Wikipedia defines a sample as: a subset of a population. While exploring the reason why we divide by $(n-1)$ instead of $n$ when calculating standard deviation (discussed in this question), I came ...
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### Is my work correct (easy problem, confidence intervals)

The r.v. $X$ represents the time taken by a computer in company $1$ in order to perform a certain job, and $Y$ represents the same thing but for company $2$. A sample of $n_X = 12$ computers are taken ...
### Expected number of successes from $N$ Bernoulli trials with different $p$
Suppose I have N probabilities $(p_1, p_2,...,p_N)$ that represent the chance that each that a corresponding test was passed. How do I apply the Bernoulli distribution to determine the expected number ...
### Why is $\frac{\sum^n_{i=1}(X_i-\bar{X})^2}{\sigma^2}$chi-square distributed with $n-1$ degrees of freedom?
On Wikipedia, it says If $X_{i};i=1,\ldots ,n$ are independent normal $(\mu ,\sigma ^{2})$ random variables, the statistic $$\frac{\sum\limits^n_{i=1}(X_i-\bar{X})^2}{\sigma^2}$$ ...