Use the definition in Expression (3.13) to prove that V(aX + b) = ?2. ?2x [ Hint: With h(X) = aX + b, E[h(X)] = aµ = b where µ = E(X).] Reference Expression (3.13 The variance of h(X) is the expected value of the squared difference between h(X) and its expected value: When h(X) = aX + b, a linear function, Substituting this into (3.13) gives a simple relationship between V[h(X)] and V(X):

Problem 41E Answer: Step1: We have the variance of h(X) is the expected value of the squared difference between h(X) and its expected value: V [h(x)]= 2 = {h(x) E[h(x)]} • P(x) ……….(1) h(x) x When h(X) = aX + b, a linear function, h(x) E[h(x)] = ax + b (a + b) = a(x ) Substituting this into (1) gives a simple relationship between V[h(X)] and V(X): We need to prove V (aX + b) = a • 2 x Step2: Let us assume that Y = aX + b W.k.t, V [h(x)]= 2 = {h(x) E[h(x)]} •...