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
show that the conditional distribution of (x1, x2) given x3 has
mean vector [µ1 + ρ2(x3 − µ3), µ2]
T and covariance matrix:
(b) If x ∼ Np(µ, Σ) random variables and QΣQT
(q × q) is nonsingular,
then, given Qx = q, show that the conditional distribution
of X is normal with µ + ΣQT(QΣQT)−1(q − Qµ) and covariance
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 population.
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:
(a) Perform a principal component analysis using the covariance matrix.
Can the data be effectively summarised in fewer than six
dimensions?
(b) Perform a principal component analysis using the correlation matrix.
(c) Comment on the similarities and differences between the two analyses.
3. (15 points) Consider the data in file data3.txt. Cluster the data using
K-means method. Comment on your findings.
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