University of California, Los Angeles Department of Statistics
Statistics 100B Instructor: Nicolas Christou Multivariate normal – practice problems
EXERCISE 1
Let (X1,Y1),…,(Xn,Yn), be a random sample from a bivariate normal distribution with parameters μ1, μ2, σ12, σ2, ρ. (Note: (X1, Y1), . . . , (Xn, Yn) are independent). What is the jointdistributionof(X ̄,Y ̄)? Hint: Findthejointmomentgeneratingfunctionof(X ̄,Y ̄)and compare it to the joint moment generating function of multivariate normal distribution.
EXERCISE 2
Answer the following questions:
a. Let X1, X2, X3 be i.i.d. random variables N(0, 1). Show that Y1 = X1 + δX3 and
Y2 = X2 + δX3 have bivariate normal distribution. Find the value of δ so that the
correlation coefficient between Y1 and Y2 is ρ = 1 . 2
b. Let X and Y follow the bivariate normal distribution with parameters μ1, μ2, σ12, σ2, ρ. Show that W = X −μ1 and Q = (Y −μ2)−ρσ2(X −μ1) are independent normal
random variables.
EXERCISE 3
σ1
Answer the following questions:
a. Let X1 and X2 be two independent normal random variables with mean zero and
variance 1. Show that the vector Z = (Z1, Z2)′, where Z1 =μ1+σ1X1
Z =μ+ρσX+σ1−ρ2X 222122
follows the bivariate normal distribution with parameters μ1, μ2, σ12, σ2, ρ. X Y
b. Suppose 1 ∼ N2(μ,Σ). Consider the vector 1 , where Yi = eXi,i = 1,2. X2 Y2
Find EY13 and covariance between Y13, Y23.