My notes claim the following:
The variance of a random variable $X$ is
$$Var[X] = E[(X – E[X])^2]$$
$\dots$
For any random variable $X$,
$Var[X] = E[X^2] – (E[X])^2$
I'm wondering how it is that $Var[X] = E[(X – E[X])^2] = E[X^2] – (E[X])^2$?
We have that
$$Var[X] = E[(X – E[X])^2] = E[X^2] – 2XE[X] + (E[X])^2 \not= E[X^2] – (E[X])^2$$
I would greatly appreciate it if people could please take the time to clarify this.
Best Answer
$$Var[X] = E[(X - E[X])^2] = E[X^2] - 2E[XE[X]] + (E[X])^2 =E[X^2] - 2E[X]E[X] + (E[X])^2 $$ that is $$Var[X] =E[X^2] - 2(E[X])^2 + (E[X])^2 $$