4.2. MOMENT GENERATING FUNCTIONS 133
Proof.
φZ(t) =E[
etX]
=
1
√
2 πσ^2∫∞
−∞etxe−(x−μ)(^2) /(2σ (^2) )
dx
=
1
√
2 πσ^2∫∞
−∞exp(
−(x^2 − 2 μx+μ^2 − 2 σ^2 tx)
2 σ^2)
dxNow by the technique of completing the square:
x^2 − 2 μx+μ^2 − 2 σ^2 tx=x^2 −2(μ+σ^2 t)x+μ^2
= (x−(μ+σ^2 t))^2 −(μ+σ^2 t)^2 +μ^2
= (x−(μ+σ^2 t))^2 −σ^4 t^2 − 2 μσ^2 tSo returning to the calculation of the m.g.f.
φZ(t) =1
√
2 πσ^2∫∞
−∞exp(
−((x−(μ+σ^2 t))^2 −σ^4 t^2 − 2 μσ^2 t)
2 σ^2)
dx=
1
√
2 πσ^2exp(
σ^4 t^2 + 2μσ^2 t
2 σ^2)∫∞
−∞exp(
−(x−(μ+σ^2 t))^2
2 σ^2)
dx= exp(
σ^4 t^2 + 2μσ^2 t
2 σ^2)
= exp(
μt+σ^2 t^2 / 2)
Theorem 11.IfZ 1 ∼N(μ 1 ,σ^21 ), and Z 2 ∼N(μ 2 ,σ^22 )andZ 1 andZ 2 are
independent, thenZ 1 +Z 2 ∼ N(μ 1 +μ 2 ,σ^21 +σ^22 ). In words, the sum of
independent normal random variables is again a normal random variable
whose mean is the sum of the means, and whose variance is the sum of the
variances.
Proof. We compute the moment generating function of the sum using our
theorem about sums of independent random variables. Then we recognize the
result as the moment generating function of the appropriate normal random
variable.
φZ 1 +Z 2 (t) =φZ 1 (t)φZ 2 (t)
= exp(μ 1 t+σ 12 t^2 /2) exp(μ 2 t+σ 22 t^2 /2)
= exp((μ 1 +μ 2 )t+ (σ 12 +σ 22 )t^2 /2)