MATH 556 - PRACTICE EXAM QUESTIONS II
1. Suppose that X1 , X2 , . are i.i.d Cauchy random variables with pdf
fX (x) =
1 1
1 + x2
xR
and characteristic function CX (t) = expcfw_|t|.
(a) Find the distribution of the random variable Tn dened by
n
Tn =
Xi .
MATH 556 - ASSIGNMENT 1 SOLUTIONS
1. For the discrete variables concerned
(a) As
x=0 y=0
(x + y)y
x
y
y
=
=
x
+
x!
y!
x!
y!
(y 1)!
x=0
y=0
x=0
y=0
y=1
x
y
x
y
=
x
=
xe + e
+
x!
y!
y!
x!
x+y
(x + y)
x!y!
x
x=0
y=0
= e
x=1
x=0
y=0
x
+
(x 1)!
= e (
MATH 556 - ASSIGNMENT 1
To be handed in not later than 5pm, 28th September 2006.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1. Suppose X and Y are discrete random variables having joint pmf given by
fX,Y (x,
MATH 556 - ASSIGNMENT 3
To be handed in not later than 5pm, 16th November 2006.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1 Consider the three-level hierarchical model:
LEVEL 3 : > 0, r cfw_1, 2, . . .
Fixe
MATH 556 - ASSIGNMENT 2
To be handed in not later than 5pm, 19th October 2006.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1
(a) Suppose that X is a continuous rv with pdf fX and characteristic function (cf)
MATH 556 - ASSIGNMENT 3 SOLUTIONS
1
(a) By iterated expectation, using the formula sheet to quote expectations for Gamma and Poisson
Ef [N ] + r/2
N + r/2
+ r/2
= N
=
= 2 + r
1/2
1/2
1/2
EfX [X] = EfN [EfX|N [X|N = n] = EfN
3 M ARKS
(b) By the same metho
MATH 556 - ASSIGNMENT 4 SOLUTIONS
1 For t > 0, and constant k > 0
P [X t] = P [X + k t + k] P [(X + k)2 (t + k)2 ]
EfX [(X + k)2 ]
(t + k)2
by the Chebychev Lemma. Now if k = 2 /t, then
P [X t]
EfX [(X + 2 /t)2 ]
Ef [(tX + 2 )2 ]
= X 2
(t + 2 /t)2
(t +
MATH 556 - MID-TERM SOLUTIONS
1.
(a) (i) From rst principles (univariate transformation theorem also acceptable): for 0 < x < 1
2
2
arcsin x = arcsin x
FX (x) = P [X x] = P [sin(U/2) x] = P U
and zero otherwise, as the sine function is monotonic increasi
MATH 556 - PRACTICE EXAM QUESTIONS II SOLUTIONS
1.
(a) Using properties of cfs, we have
CTn (t) = e|t|
n
= e|nt|
Now using the scale transformation result for mgfs/cfs (given on Formula Sheet), we have
that if V = U , then
CV (t) = CU (t)
we deduce that,
MATH 556 - MID-TERM EXAMINATION
Marks can be obtained by answering all questions. All questions carry equal marks.
1.
(a) Suppose that U is a continuous random variable, and U U nif orm(0, 1). Let random
variable X be dened in terms of U by
X = sin(U/2).
MATH 556 - PRACTICE EXAM QUESTIONS
1. Due to the symmetry of form, this joint pdf factorizes simply as
fX,Y (x, y) =
x
c1 exp
2
y
c1 exp
2
= fX (x)fY (y)
x, y > 0
and hence the variables are independent. Now
exp
0
x
dx = 2
2
so therefore c1 = 1 , and h
MATH 556 - ASSIGNMENT 2 SOLUTIONS
1
(a) (i) By direct calculation
CX (t) = EfX [eitX ] =
eitx expcfw_x ex dx =
expcfw_(1 it)x ex dx
=
y it ey dx = (1 it)
0
after setting y = ex .
Note that the Gamma function notation usage here is legitimate; the Gamma
MATH 556 - ASSIGNMENT 4
To be handed in not later than 5pm, 30th November 2006.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1 Suppose that X has expectation zero, and nite variance 2 . Prove that, for t > 0,
MATH 556 - PRACTICE EXAM QUESTIONS
1. The joint pdf for continuous random variables X, Y with ranges X Y R+ is given by
1
fX,Y (x, y) = c1 exp (x + y)
2
x, y > 0
and zero otherwise, for some normalizing constant c1 .
Consider continuous random variable U