# the expectation of the sum of yi minus ybar squared

• Nov 5th 2013, 12:16 AM
kingsolomonsgrave
the expectation of the sum of yi minus ybar squared
In general what is the expectation of the sum of yi minus ybar squared

$\displaystyle E(\sum( y_i - \bar{y})^2) = E(\sum( y_i^2 - 2\bar{y}y_i + \bar{y}^2)) =$

$\displaystyle E(\sum y_i^2 -\sum \bar{y}^2)) = \sum E(y_i^2) -\sum E(\bar{y}^2)$

is this the same as the sum of of VAR(Y)?
• Nov 5th 2013, 03:50 PM
chiro
Re: the expectation of the sum of yi minus ybar squared
Hey kingsolomonsgrave.

What do you mean by sum of Var[Y]? Can you write it out in terms of sigma notation and expectation/variance of random variables?
• Nov 6th 2013, 11:15 AM
kingsolomonsgrave
Re: the expectation of the sum of yi minus ybar squared
var(Y) in terms of expectation is $\displaystyle E (Y_i - \bar{Y)}^2 = E(Y^2)-[E(y)]^2$

and

$\displaystyle E(\sum (Y_i -\bar{Y})^2) = E[\sum(Y_i^2 -2Y_i\bar{Y} +\bar{Y})]$

$\displaystyle =\sum[E(Y_i ^2) - [E(\bar(Y))^2] = \sum(VAR(Y)) = nVAR(Y)$
• Nov 6th 2013, 01:49 PM
chiro
Re: the expectation of the sum of yi minus ybar squared
This looks right and remember also that expectation is linear so if you have a sigma inside the expectation, you can take it outside the expectation as well.