Let X be a random variable with CDF Fx Show that EXc2 is min

Let X be a random variable with CDF F(x).

Show that E[(X-c)^2] is minimized by the value c=E(X)

Solution

This can be showed using the formula of Variance :

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

= E(X^2 ) -2 E(X)^ 2 + E(X) ^2

= sum x^2 p(x) -2 E(X) sum xp(x) + E(X)^ 2 sum p(x)

= sum (x^2 -2x E(X)+ E(X) ^2 ) p(x)

= sum (x- E(X)) ^2 p(x)

= E((X-E(X)) ^2 )

Comparing above equation with E[(X-c)^2] ,

We get ,

c = E(X)

Proved