Probability , expectation, variance, cross-term vani

[binbagsss]

Homework Statement

I have a variable $s_i$ with probability distribution $w(s_i)$
$(\Delta(s_i))^2$ denotes the variance $=<(s-<s>)^2>=<s^2>-<s>^2$
I want to show $\sum\limits_{i\neq j} <\Delta s_i> < \Delta s_j> =0$

where $< >$ denote expectation

My book has:

$<\Delta s_i> =\int ds_i w(s_i)(s_i-<s_i>)=0$

I don't really understand this so the first term gives $<s_i>$ that's fine which would obviously cancel with a $<s_i>$ but isn't the second term $E(<s_i>)$ not $<s_i>$ so how do they cancel?

Many thanks in advance.

Homework Equations

see above

The Attempt at a Solution

[/B]
I don't really understand this so the first term gives $<s_i>$ that's fine which would obviously cancel with a $<s_i>$ but isn't the second term $E(<s_i>)$ not $<s_i>$ so how do they cancel?

Many thanks in advance.

 

[Ray Vickson]

Homework Statement

I have a variable $s_i$ with probability distribution $w(s_i)$
$(\Delta(s_i))^2$ denotes the variance $=<(s-<s>)^2>=<s^2>-<s>^2$
I want to show $\sum\limits_{i\neq j} <\Delta s_i> < \Delta s_j> =0$

where $< >$ denote expectation

My book has:

$<\Delta s_i> =\int ds_i w(s_i)(s_i-<s_i>)=0$

I don't really understand this so the first term gives $<s_i>$ that's fine which would obviously cancel with a $<s_i>$ but isn't the second term $E(<s_i>)$ not $<s_i>$ so how do they cancel?

Many thanks in advance.

Homework Equations

see above

The Attempt at a Solution

[/B]
I don't really understand this so the first term gives $<s_i>$ that's fine which would obviously cancel with a $<s_i>$ but isn't the second term $E(<s_i>)$ not $<s_i>$ so how do they cancel?

Many thanks in advance.

$\langle s_i \rangle$ is just a number, so $\int w(s_i) \langle s_i \rangle \, ds_i = \langle s_i \rangle \int w(s_i) \, ds_i$, and that last integral equals 1.