Question

Recall that the variance of a random variable is defined as

Var[X]=E[(X−μ)2], where μ = E[X]. Use the properties of expectation to show that we can rewrite the variance of a random variable X as Var [X]=E[X^2]−(E[X])^2

Problem 3. (1 point) Recall that the variance of a random variable is defined as Var X-E(X-μ)21, where μ= E[X]. Use the properties of expectation to show that we can rewrite the variance of a random variable X as u hare i- ElX)L a

Answer 1

Var(X) =E(X-E(X))² =E(X²+(E(X))²-2XE(X)) =E(X²)+E(((E(X))²-2*E(x*E(X))

=E(X²)+(E(X))²-2*E(X)*E(X) =E(X²)+(E(X))²-2*(E(X))²

Var(X)=E(X²)-(E(X))²