Question

Let at XW) =andom variable. Prove III. VARIANCE PROOFS (SINGLE RANDOM VARIABLE) Let S be a sample space. Let a, b, c be real numbers. a) Let X(W) = c for all w ES. Prove that Var(X) = 0. b) Let Y be a random variable. Prove that Var(Y + b) = Var(Y). c) Let Z be a random variable. Prove that Var(az) = a-Var(Z). d) Let W be a random variable. Use parts (b)-(c) to prove that Var(aW+b) = aVar(W).

Answer 1

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4) 3) X(W) = c WES X is constant random van'able and therefore E(X) = F(C) = c E (X)= ECC) Therefore variance of x is Var (x) = E Cx2) - (ECX))2 = c2- (2 → Var(x) = 0 b) Now, 6 be a any constant → E (x+b) = =) ECY+6) = E(Y) + b E(Cy+633) - E (y2+264 +63) = E(72) +26 ECY) +62 There fore, Var(y+b) = EC Y+b)2- CE CY+b)) 2 = E CY+6)2- ( E(Y) +b)2 = E(42) + 2b E(4) +62 - (E(Y))2 - 2-6 E14) - 62 = E(42) - CECY))2

.:. Vary+b) = ar Cy+b) = Ely E (42) - CECY?)) = varly) 3) Var Cy+b) = Var (9) constant C) Now, z is random van'able :: Var (az) = E(( az)3) - ( E(az))2 žais = E(Q2 22) -( = 02 E122) - (Q E(z))} 3(62) = Q EC2) = Q2 ( E(22) - (E(z)) - a2 Var(2) var(az) = az variz) d) Now, W be a random van'able var cawt b) = EC cakl+b)2 var cald+b) = varcaw) ſ from part (6) = azvar (W) { from part (a) Varc ab +b) = Q? Var(w) Hence proved
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