MATH 556 - ASSIGNMENT 3
To be handed in not later than 5pm, 15th November 2007.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1
(a) State whether each of the following functions denes an Exponential Family dist
MATH 556 - ASSIGNMENT 1
To be handed in not later than 5pm, 20th September 2007.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1. Suppose that X is a discrete random variable with pmf fX specied by
fX (x) =
k
x
MATH 556 - A SSIGNMENT 3
S OLUTIONS
1
(a) (i) This is not an Exponential Family distribution; the support is parameter dependent.
1 M ARK
(ii) This is an EF distribution with k = 1:
f (x|) =
where
Icfw_1,2,3,. (x)
1
expcfw_x log = h(x)c() expcfw_w()t(x)
MATH 556 - A SSIGNMENT 4
S OLUTIONS
1 By properties of the Gamma distribution, we can write
Vn =
(n 1)s2
n
2 Gamma
n1
2
n1 1
,
2
2
d
n1
= Vn =
Xi
i=1
d
where Xi 2 Gamma (1/2, 1/2), and the symbol = indicates equality in distribution (that
1
is, the left
MATH 556 - A SSIGNMENT 1
S OLUTIONS
1. We have, for x = 1, 2, . . .
x
FX (x) =
x
t=1
but
k
t(t + 1)
fX (t) =
t=1
1
1
1
=
t(t + 1)
t
t+1
so, in fact
x
FX (x) = k
t=1
1
1
k
kx
=k
=
t
t+1
x+1
x+1
as the sum telescopes. Noting that we must have FX (x) 1 as x
MATH 556 - ASSIGNMENT 4
To be handed in not later than 5pm, 29th November 2007.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
In the following questions, use the key stochastic convergence concepts for a sequen
MATH 556 - ASSIGNMENT 2
To be handed in not later than 5pm, 11th October 2007.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1. Suppose that X1 and X2 are continuous random variables, with joint pdf specied as
MATH 556 - ASSIGNMENT 1: SOLUTIONS
1 The distribution of any discrete random variable, X, can be written as a linear combination of a countable number of point mass measures
PX (B) =
i=1
pi xi (B)
where x1 , x2 , . . . are the countable set of values at w
MATH 556 - MID-TERM SOLUTIONS 2008
1.
(a) From rst principles (univariate transformation theorem also acceptable): for z (0, 1/4)
FZ (z) = PZ [Z z] = PX [X(1 X) z] = PX [X x1 (z) X x2 (z)]
where x1 (z) and x2 (z) are the roots of the quadratic x2 x + z =
MATH 556 - MID-TERM EXAMINATION 2006
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
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
M ATH 556 - P RACTICE E XAM Q UESTIONS : S OLUTIONS
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 the
MATH 556 - ASSIGNMENT 3
To be handed in not later than 5pm, 11th November 2008.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1 Suppose that two random variables X and Y (dened on the same probability space (,