Question

. Show that E(X − c)
2
is minimized by taking
c = E(X).

Answer 1

$$E[(X-c)^2] = E[X^2-2cX+c^2]=E[X^2]-2cE[X]+c^2$$
now consider that to minimize this in terms of \(c\) we see: $$c^2-2E[X]c+E[X^2]=(c-E[X])^2+E[X]^2-E[X^2]$$ by completing the square

Answer 2

this quadratic in \(c\) is minimized when \(c=E[X]\), which coincides with its vertex. you can also determine this by differentiating w.r.t. c: $$2(c-E[X])=0\\\implies c=E[X]$$since \(E[X]^2-E[X^2]\) is a constant