ST2352 Problem Set 4 2012
Multivariate Normal Distribution
Several drawn from Tijms Ch 12. The Wiki pages http://en.wikipedia.org/wiki/Inverse_matrix and
http://en.wikipedia.org/wiki/Multivariate_normal_distribution are useful
Prove that aX+bY has a scalar normal distribution for any constants a and b, if (X,Y) has a bivariate
normal pdf. (Note that the proof is trivial if (X,Y) have been constructively defined as combinations
of independent scalar normal rvs.)
1. The rate of return on stocks A and B has a bivariate normal distributions, with Expected Values
0.08% and 0.12%, resp), SDs (0.05%, 0.15%) and correlation -0.50. What are the probabilities
that the return on A exceeds that on B? that the average rate of return in larger than 0.11%?
2. It is known that that the rate of return on A (in Q2) is 0.10. What is the conditional probability
distribution of the distribution of B? What is the conditional probability distribution of the
average of A and B?
3. A student is going to predict the rate of return on A (in Q2) by using a linear function of the
return on B; ie      B . The best such function will have E  A  Aˆ   0 and minimise


Var  A  Aˆ  . Express this in terms of ,, and the parameters of the joint distribution of A and
B. Hence find the values of ,. This is the Best Linear Unbiased Predictor BLUP of A. Contrast
with Q3.
4. If (X,Y) have a bivariate normal pdf with Var(X)=Var(Y), show that (X+Y) and (X-Y) have
independent normal distributions.
5. The annual rates of return on three stocks A, B and C have a trivariate normal distribution. The
expected values are 7.5%, 10%, 20%, resp, and SDs 7%,12%18% resp; these SDs are known as
‘risks’. The correlations are 0.7, -0.5, -0.3, for AB, AC and BC resp. An investor has €100,000 in
cash to allocate to the funds. Any unallocated funds are invested in an asset D that offers 5% and
is riskless (with SD = 0).
a. If he invests €20,000, €20,000 and €40,000 in A,B and C, which defines a ‘portfolio, what
is the expected rate of return? what is its SD?
b. Find a portfolio with smaller risk than in part (a) but with expected return greater than in
(a).
c. If the proportions invested in each are  pA , pB , pC   p (summing to 1), is there a p
that maximises return? that minimises risk?
6. What is the BLUP of A based on C? What is the BLUP of B based on C? What is the correlation
between these BLUPs. What is the BLUP of A based on both B and C.
7.
A multivariate Normal random variable Y, a vector of length n, has vector mean  and variance
matrix .
a.
Such a variable Y may be simulated by forming a linear combination Y 
independent random variables
  AZ of n
Zi , such as below, each with distribution N(0,1). Explain,
illustrating by using  with elements (1, 0, 2) and the matrix A shown.
A
b.
Values drawn from N(0,1)
1
0
0
-0.179
0.202
-0.392
-1.446
-0.807
1
2
0
-0.407
-1.444
-1.093
0.217
-0.922
1
1
3
0.491
0.490
-1.255
-0.986
-0.804
What is the variance matrix  here. What is the joint pdf of Y? What is the marginal
distribution of Y1? What is its pdf? Illustrate using the example in part a.
c.
What, for generic (, ) is the pdf of Y1+Y2? Illustrate using the example in part a.
d.
If (Ya,Yb) is a partition, state and explain the conditional distribution of Ya given Yb. Illustrate
using the example above, where the partition is a = {1,2}, b={3} and Y3 =0.


8. If the distribution of Y is MVN, then its pdf fY ( y ) can be written as  exp  12 h  y;  , Q 
where h  y;  , Q    y    Q  y    and Q is the inverse of the variance matrix  . If
T
Y 
 

Y   a  ,    a  ,    aa
 Yb 
 b 
  ba
 ab 
 Qaa Qab 
,Q  
 expand the function h  y;  , Q  in
 bb 
 Qba Qbb 
terms of the components of  y;  , Q  . The expansion can be re-expressed as
 ya  Byb T Raa  ya  Byb    yb  b T  bb1  ya  b  for suitable B, Raa . Identify the terms
B, Raa , Rbb in the expansion. Hence give an expression for the conditional pdf of Ya , given that
Yb  yb . Contrast with the conditional distribution information at
http://en.wikipedia.org/wiki/Multivariate_normal_distribution.
9. Relate the results of Q8 to Q7d and to Q2.