How do you Derive variance formula?

Answer 1

var(x) = E[(x - E(x))2]

= E[(x - E(x)) (x - E(x))] <-------------Expand into brackets

= E[x2 - xE(x) - xE(x) + (E(x))2] <---Simplify

= E[x2 - 2xE(x) + (E(x))2]

= E(x2) + E[-2xE(x)] + E[(E(x))2]

= E(x2) - 2E[xE(x)] + E[(E(x))2] <---Bring (-2) constant outside

= E(x2) - 2E(x)E[E(x)] + E[(E(x))2] <--- E[xE(x)] = E(x)E(x)

= E(x2) - 2E(x)E(x) + [E(x)]2 <----------E[E(x)] = E(x)

= E(x2) - 2[E(x)]2 + [E(x)]2

var(x) = E(x2) - [E(x)]2