# 统计代写 | STAT3017/STAT7017 Assignment 2

Question 1 [5 marks]

Let Y be a random vector with multivariate Gaussian distribution Np(0; ). Show that if
rank( ) = p then
Y 0
1
Y  2
(p);
where 2(p) denotes the chi-squared distribution with p degrees of freedom.

Question 2 [5 marks]

Calculate the following integral
1
4i
I
jzj=2
z3
e1=z
dz:

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
conclude?

(c)  Through a simulation study1, reproduce Figure 6 of the paper for various choices of
p = 2; 10; 50; 100. What can you conclude?

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