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Suppose $X_1,...,X_n$ is a random sample from a distribution having $N(\mu, \sigma^2)$. What is the conceptual difference between:

$$ \frac1{n} \sum_{i=1}^n (X_i - \bar{X})^2$$

and

$$ \frac1{n} \sum_{i=1}^n (X_i - \mu)^2 ?$$

And why, when multiplied by $n$ and divided by $\sigma ^2$, the first becomes a $\chi^2 (n-1)$ while the second $\chi^2 (n)$?

George
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  • This might help: https://en.wikipedia.org/wiki/Degrees_of_freedom_(statistics) –  Aug 20 '15 at 19:19
  • Related to $\displaystyle E\left[\frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2\right] = E\left[\frac{1}{n} \sum_{i=1}^n (X_i - \mu)^2\right] $ – Henry Aug 20 '15 at 19:20
  • it is about what is random and what is not..$\mu$ is not random and $\bar{X}$ is only asymptotically non-random. – Seyhmus Güngören Aug 20 '15 at 19:32

0 Answers0