(1.1)
Za = αZx + (1-α)Zy
E[Za] = αE[Zx] + (1-α)[Zy] = αμx + (1-α)μy
(1.2)
V[Za] = α²V[Zx] + 2α(1-α)cov[Zx,Zy]+(1-α)²V[Zy] = α²σ²x + 2α(1-α)ρσxσy+(1-α)²σ²y
(1.3)
V[Za] = (σ²x - 2σxσy + σ²y)α² + 2(σxσy - σ²y)α + σ²y
αで微分 2 (σ²x - 2σxσy + σ²y)α + 2(σxσy - σ²y) = 0
∴ α = -(σxσy - σ²y) / (σ²x - 2σxσy + σ²y)
(1.4)
V[Za]=0 が解をもつとき D/4≧0
(σxσy - σ²y)² - (σ²x - 2ρσxσy + σ²y)σ²y = 2σ²y (-1+ρ)σxσy≧0
-1+ρ≧0 ρ≧1 ∴ρ=1