Show that varaXEaXEaX2 can be simplified to varaXa2varX Solu

Show that var(aX)=E(aX-E(aX))^2 can be simplified to var(aX)=a^2var(X) .

Solution

var(aX)=E(aX-E(aX))^2

Expand

(aX-E(aX))^2 = a²x²+2aXE(aX)-{E(aX)}²

⁼ a²x²+2a²XE(X) - {aE(X)}² Using property of Expectation

Thus

var(aX)=E(aX-E(aX))^2

=E(a²x^2-2a²XE(X) + {aE(X)}² )   

E(X) is a constant here also a is a constant, hence can be taken out

= a²E(X²) - 2a²E(X)E(X) +a²{E(X)}² )   

Grouping II and III term

= a²E(X²) -a²{E(X)}²

Taking a square out

= a²(E(X²) -{E(X)}²⁾

= a²Var(X)