ST5222: Advanced Topics in Applied Statistics
Midterm 1
Dealine for submission midnight 9th of October, 2019.
1. (10 points)
(a) Suppose (x1, x2, x3) follow a multivariate normal distribution with
mean (µ1, µ2, µ3) and covariance matrix
Σ =
 1 ρ ρ2ρ 1 0
ρ2 0 1

show that the conditional distribution of (x1, x2) given x3 has
mean vector [µ1 + ρ2(x3 − µ3), µ2]T and covariance matrix:[
1− ρ4 ρ
ρ 1
]
.
(b) If x ∼ Np(µ,Σ) random variables and QΣQT (q × q) is non-
singular, then, given Qx = q, show that the conditional distri-
bution of X is normal with µ + ΣQT (QΣQT )−1(q − Qµ) and co-
variance matrix Σ− ΣQT (QΣQT )−1QΣ.
2. (15 points) A naturalist for the Alaska Fish and Game Department
studies grizzly bears with the goal of maintaining a healthy popula-
tion. Measurements on n = 61 bears provided the following summary
statistics:
1
Variable Weight Body Neck Girth Head Head
(kg) length (cm) (cm) length width
(cm) (cm) (cm)
Sample
mean x¯ 95.52 164.38 55.69 93.39 17.98 31.13
Covariance matrix:
S =

3266.46 1343.97 731.54 1175.50 162.68 238.37
1343.97 721.91 324.25 537.35 80.17 117.73
731.54 324.25 179.28 281.17 39.15 56.80
1175.50 537.35 281.17 474.98 63.73 94.85
162.68 80.17 39.15 63.73 9.95 13.88
238.37 117.73 56.80 94.85 13.88 21.26

(a) Perform a principal component analysis using the covariance ma-
trix. Can the data be effectively summarised in fewer than six
dimensions?
(b) Perform a principal component analysis using the correlation ma-
trix.
(c) Comment on the similarities and differences between the two anal-
yses.
3. (15 points) Consider the data in file data3.txt. Cluster the data using
K-means method. Comment on your findings.
2