Georgia Institute Of Technology

MATH 4261

Please write your solutions on the scratch paper that was provided with the exam.
Explain all of your solutions/answers to receive a full score.
Problem 1
Consider random variables X ~ N (u1,0%) and Y ~ N (u2,03) for some u1,us € R and 0%, 05 > 0.
2 05
(a) (1 point) Let a variance-covariance matrix of Z := (X,Y)T be Var(Z) = (;'15 2) .
. 05
Find the expected value E(Z).
(b) (2 points) Argue whether or not vector Z has a multivariate normal distribution.
(c) (2 points) Find the correlation coefficient p(10X — 1,5 —Y).
(d) (2 points) Now assume that X and Y are independent from each other. Find p(10X —1,5—-Y).