Hello I need assistance with B22 please thank you very much

Hello, I need assistance with B.22 please thank you very much.

B.22. Show that E(X - k)^2 = var (X) + [E(X) - k]^2. For what value of k will E(X - k)^2 be minimum? And what is that value of k?

Solution

R.H.S = Var(x) + [E(X) - k]² = Var(X) + [E(X)² + k² - 2kE(X)] = E(X²) - E(X)² +E(X)² + k² - 2kE(X)

= E(X²)+ k² - 2kE(X) = E(X - k)² = L.H.S

for k = E(X) we will have minimum value of E(X - k)² which will become the varaince formula