TITLE: Expected value equal to expected value of expected value squared?
QUESTION [5 upvotes]: I am trying to rederive the canonical expression for variance $V[X] = E[X^2] - E[X]^2$. 
What I don't understand is the second line. If you multiply E( ... ) by the last term, wouldn't you get E(E[X]^2)? Is $E(E[X]^2) = E[X]^2$ ?
 
$E(E[X]^2) = E[X]^2$

REPLY [6 votes]: Remember that the expected value is a measure of some central tendency, the average.  When discussing the average, we are talking about a particular number, a constant.  From here, we can see that squaring the expected value is just squaring a constant yielding another constant.  Thus if 
$E[X]=\mu$ for some random variable $X$, then $E[X]^2=\mu^2$.  Now what is 
$E[E[X]^2]$
$$E[E[X]^2]=E[\mu^2]=\mu^2=E[X]^2$$