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Statistics

The probability density of obtaining a sample mean \bar x_n is the product of probabilities of obtaining arbitrary values x_1, x_2,\dots,x_n with the constraint that the mean of the set \{x_i\} is \bar x_n :

p_{\overline X_n}(x) = \int p_X^{\phantom X}(x_1)\cdots \int p_X^{\phantom X}(x_n)\ \delta\!\left(x - \frac{x_1+x_2+\dots+x_n}{n}\right)dx_n \cdots dx_1

And in particular we are interested in its variance \mathrm{var}(\overline X_n) .