1. Problem 12.6 The random variables Z1 and Z2 are independent and N(0,1) distributed. Define the random variables X1 and X2 by X1 = 1

1. Problem 12.6 The random variables Z1 and Z2 are independent and N(0,1)

distributed. Define the random variables X1 and X2 by X1 = μ1 + σ1Z1 and X2 =μ2+σ2ρZ1 +σ2 1 −ρ2Z2, where μ1, μ2, σ1, σ2 and ρ are constants withσ1 > 0,σ2 > 0and−1 <ρ<1.Provethat(X1, X2)hasabivariatenormal distribution with parameters (μ1,μ2,σ2 1,σ2 2,ρ).

12.1 Bivariate normal distribution 335