University of California, Los Angeles Department of Statistics
Instructor: Nicolas Christou
Quiz 3
a. If Y1 and Y2 are random variables such that X1 = Y1 + Y2 and X2 = Y1 − Y2 are independent N (0, 1) random variables, show that Y1 and Y2 have a bivariate normal distribution. Find the mean and variance covariance matrix of Y = (Y1, Y2)′.
b. Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 +X2 +X3,Y2 = X1 −X2,Y3 = X1 −X3. Find the joint pdf of Y = (Y1, Y2, Y3)′ using:
1. The method of variable transformations (Jacobian). 2. Multivariate normal distribution properties.
c. 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
Statistics 100B
Answer the following questions:
1