MAST30020
HW Solutions 1/2016
1. (i) When determining if infinitely many of the En s occurred, we can ignore
any fixed finite collection of events Ei1 , . . . , Eik (there are only finitely many
of them!). So we can ignore E1 and E2 .
If B then E3 , E5 ,

MAST30020
Tutorial Solutions 6
1. Recall that we already know that Z P ( + ), right?
Given that Z = n 0, the value of X can only be one of k = 0, 1, . . . , n. By the
definition of conditional probability (given an event), for such values of k, one has
P

MAST30020
HW Solutions 4/2016
1. (a) Start with vizualising the problem: draw the support of the distribution
of (X1 , X2 ), i.e. the triangle := cfw_(x1 , x2 ) : x1 0, |x1 | + 2|x2 | 2.
It has vertices at points (0, 1), (0, 1) and (2, 0).
Note that the s

MAST30020
HW Solutions 5/2016
1. Its all just Jensens inequality.
(a) Let g(x) := (ln x)2 , x > 0. One has
00
2
g (t) = 2(1 ln t)t
> 0, t (0, e);
< 0, t > e.
So g is convex on (0, e) and concave on (e, ). So for X e a.s., apply
Jensens inequality to g to

MAST30020
Tutorial Solutions 5
R
1. Using the formula E Z = 0 (1 FZ (t) dt that holds for any RV Z 0 (see lecture slide 71) and setting Xa = X a for brevity, one obtains, using the result of
Problem 3(c) from PS-3, that
Z
Z
+
(1 FXa+ (t) ) dt +
E |X a|

MAST30020 Probability & Statistical Inference
Problem Sheet 7
Solutions to the homework problems are to be left in the MAST30020 assignment box #200
on the ground floor in the Richard Berry Building (the boxes are located in the corridor
leading to Wilson

MAST30020
Tutorial Solutions 7
1. One has fX|Z (x|z) = f(X,Z) (x, z)/fZ (z), where, by the convolution formula, for z > 0,
Z
fZ (z) = fX+Y (z) = fX (x)fY (z x) dx
Z
Z z
x
(zx)
2 ez dx = 2 zez .
= e 1(x > 0)e
1(z x > 0) dx =
0
1 1
To find f(X,Z) , note tha

MAST30020 Probability & Statistical Inference
Problem Sheet 6
Solutions to the homework problems are to be left in the MAST30020 assignment box
#200 on the ground floor in the Richard Berry Building (the boxes are located in the
corridor leading to Wilson

MAST30020 Probability & Statistical Inference
Problem Sheet 1
Please download the Plagiarism Declaration Form from the Web (there is a link
for it on our subjects Web site) and print it out. Complete/sign/date and attach
the form at the front of your firs

MAST30020 Probability & Statistical Inference
Problem Sheet 8
Solutions to the homework problems are to be left in the MAST30020 assignment box #200
on the ground floor in the Richard Berry Building (the boxes are located in the corridor
leading to Wilson

MAST30020 Probability & Statistical Inference
Problem Sheet 5
Solutions to the homework problems are to be left in the MAST30020 assignment box
#200 on the ground floor in the Richard Berry Building (the boxes are located in the
corridor leading to Wilson

MAST30020 Probability & Statistical Inference
Problem Sheet 4
Solutions to the homework problems are to be left in the MAST30020 assignment box
#200 on the ground floor in the Richard Berry Building (the boxes are located in the
corridor leading to Wilson

MAST30020
Tutorial Solutions 3
1. (a) First plot the density f : DIY (f is a continuous piece-wise linear triangular
function, vanishing outside [0, 2], growing linearly from 0 to 1 on [0, 1] and then
decaying to 0 on [1, 2]). The DF of Pa is given by the

MAST30020 Probability for Inference
Problem Sheet 2
Solutions to the homework problems are to be left in the MAST30020 assignment box
#200 on the ground floor in the Richard Berry Building (the boxes are located in the
corridor leading to Wilson lab). Don

MAST30020 Probability & Statistical Inference
Problem Sheet 3
Solutions to the homework problems are to be left in the MAST30020 assignment box
#200 on the ground floor in the Richard Berry Building (the boxes are located in the
corridor leading to Wilson

MAST30020
HW Solutions 3/2016
1. (a) Since arcsin x isR an odd function,
R 1the integral thereof over [1, 1] is zero
and hence 1 = f (x)dx = 0 + 1 c dx = 2c. So c = 0.5.
(b) Since, using the change of variable arcsin x = (so that x = sin ) and
integrating

MAST30020 Probability & Statistical Inference
Problem Sheet 1
Please download the Plagiarism Declaration Form from the Web (there is a link
for it on our subjects Web site) and print it out. Complete/sign/date and attach
the form at the front of your firs

MAST30020 Probability & Statistical Inference
Problem Sheet 9
Solutions to the homework problems are to be left in the MAST30020 assignment box #200
on the ground floor in the Richard Berry Building (the boxes are located in the corridor
leading to Wilson