Second moment

[apricimo]

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

 

[math]

 

 

\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}

 

[/math]

 

and then they simplify to

 

[math]

 

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

 

 

[/math]

 

where did the middle term go?

Edited January 20, 2011 by apricimo

[D H]

[math]\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[/math]

[apricimo]

[math]\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[/math]

 

 

ok... got it

Edited January 20, 2011 by apricimo