Question 1 [5 marks]
Let Y be a random vector with multivariate Gaussian distribution Np(0; ). Show that if
rank( ) = p then
where 2(p) denotes the chi-squared distribution with p degrees of freedom.
Question 2 [5 marks]
Calculate the following integral
Question 3 [10 marks]
“Every minute of every data, everywhere on the planet, dozens of companies — largely
unregulated, little scrutinized — are logging the movements of tens of millions of people
with mobile phones and storing the information in gigantic data ﬁles ”, see [A].
Read and consider the paper [B] about errors in GPS movement data.
(a)  Consider the movement data in the ﬁle ‘path.txt’ that contains 100 positions over
time and store it in the variable X of dimensions 100 x 2. Plot the path, and calculate
the distance between the start of the path P and the end of the path Q.
(b)  Perform a simulation study whereby you ﬁrst assume your movement data X contains
no measurement error, then add measurement noise ” (to each measurement) drawn
from a bivariate normal with covariance = I where I is a 2 2 identity matrix
and > 0. Vary and plot the distance d(P;Q) as a function of . What can you
(c)  Through a simulation study1, reproduce Figure 6 of the paper for various choices of
p = 2; 10; 50; 100. What can you conclude?