I can't figure out why the statement below is true. I am also confused why the first statement uses square brackets but the second statement uses round brackets. Please advise.

From https://revisionmaths.com/advanced-level-maths-revision/statistics/expectation-and-variance :

The variance of a random variable tells us something about the spread of the possible values of the variable. For a discrete random variable X, the variance of X is written as Var(X).

$Var(X) = E[(X - m)^2]$ where m is the expected value E(X)

This can also be written as:

$Var(X) = E(X^2) - m^2$

  • 2
    $\begingroup$ Just google this! $\endgroup$ Nov 11 '17 at 9:22
  • $\begingroup$ See the 'Definition' section on the Wikipedia page I have linked, starting from "The expression for the variance can be expanded:" $\endgroup$ Nov 11 '17 at 9:23
  • $\begingroup$ @Mathemagical got here by googling this! $\endgroup$ Aug 23 '20 at 21:43

If $m=\mathbb E[X]$, then $$\mathbb E[(X-m)^2]=\mathbb E[X^2]-2\underbrace{\mathbb E[X]}_{=m}m+m^2=\mathbb E[X^2]-2m^2+m^2=\mathbb E[X^2]-m^2.$$

  • $\begingroup$ thanks, I was not aware of the identity $m=\mathbb E[X]$ $\endgroup$
    – nn4l
    Nov 11 '17 at 9:29
  • 1
    $\begingroup$ @nn4l: You must have been aware, because you wrote it in your question! $\endgroup$
    – Steve D
    Nov 11 '17 at 10:24

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