Question

Prove that 

(i) V(aX) = a² V(X) 

(ii) V(X + b) = V(X)

Answer 1

(i) To show V(aX) = a² V(X) 

We know V(X) = E(X² ) – [E(X²)] 

So V(aX) = [E(a² X² )] – [E(aX)]² 

= a² E(X² ) – [aE(X)]² 

= a² E(X² ) – a [E(X)]² 

= a² {{E(X² ) – [E(X)]² } 

= a² V(X)

(ii) V(X + b) = V(X) 

L.H.S. = V(X + b) = E[(X + b²) ] – {E(X + b)}² 

= E [X² + 2bX + b² ] – [E(X) + b]² 

= E(X² ) + 2bE(X) + b² – [(E(X))² + b² + 2bE(X)] 

= E(X² ) + 2bE(X) + b² – [E(X)]² – b² – 2bE(X) 

= E(X² ) – [E(X)]² 

= V(X) 

= R.H.S.