# Second moment

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In a book I am reading they list these steps and I don't understand the simplification... BTW u is a random variable and bar over u is the average for notation clearification

$\begin{array}{l} \Delta u = u - \overline u \\ \overline {\Delta u} = \overline {(u - \overline u )} = \overline u - \overline u = 0 \\ \overline { < \Delta u > ^2 } = \sum\limits_{i = 1}^M {P(u_i } )(u_i - \overline u )^2 > 0 \\ \overline {(u_i - \overline u )^2 } = \overline {(u^2 - 2u\overline u + \overline u ^2 )} = \overline {u^2 } - 2\overline {uu} + \overline u ^2 \\ \end{array}$

and then they simplify to

$\overline {(u_i - \overline u )^2 } = \overline {u^2 } - \overline u ^2$

where did the middle term go?

Edited by apricimo

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\aligned \overline{(u-\bar u)^2} &= \overline{u^2} - 2\bar u\,\bar u + \bar u^2 \\ &= \overline{u^2} - 2 \bar u^2 + \bar u ^2 \\ &= \overline{u^2} -\bar u^2\endaligned

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\aligned \overline{(u-\bar u)^2} &= \overline{u^2} - 2\bar u\,\bar u + \bar u^2 \\ &= \overline{u^2} - 2 \bar u^2 + \bar u ^2 \\ &= \overline{u^2} -\bar u^2\endaligned

ok... got it

Edited by apricimo

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