School

University of New South Wales **We aren't endorsed by this school

Course

MATH 2070

Subject

Mathematics

Date

Oct 28, 2023

Pages

1

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UNIVERSITY OF SYDNEY
School of Mathematics and Statistics
MATH2070/2970 Optimisation and Financial Mathematics
Advanced questions are marked
A
.
Week 8
Question 1.
Let
X
be the sum of the numbers showing when two dice are thrown. Give the probability
distribution of X and calculate its mean and variance.
Question 2
(
A
)
.
The binomial distribution is a discrete distribution defined by the probability function
b
(
k
;
n, p
) =
n
k
p
k
q
n
−
k
;
0
≤
k
≤
n
and represents the probability of
k
successes in
n
independent trials when
p
is the probability of a success
in a single trial and
q
= 1
−
p
.
Using direct computations show that
E
[
X
] =
np
and
Var
(
X
) =
npq
, where
Var
(
X
)
denotes the
variance of
X
.
Question 3.
The yield
Y
of a zero-coupon bond is assumed to be a continuous random variable, expo-
nentially distributed with p.d.f. (probability density function) given by
f
(
y
) =
1
µ
e
−
y/µ
;
y
≥
0
.
Find the expected value and variance of the yield.
Question 4.
Given that the continuous compound rate
R
is normally distributed with mean
µ
and variance
σ
2
, show that the expected discount factor is
E
[
e
−
Rt
] =
e
−
(
µ
−
σ
2
t/
2)
t
.
Question 5.
Let
X
be a random variable uniformly distributed on the interval
[
a, b
]
(with
a < b
). Compute
the cumulative distribution function, expectation and variance of
X
. Recall that the p.d.f. of the uniform
distribution is
f
(
x
) =
(
1
b
−
a
if
x
∈
[
a, b
]
0
if
x /
∈
[
a, b
]
.
Question 6.
Let
X
be a random variable. Find the minimum of
γ
(
t
) =
E
[(
X
−
t
)
2
]
. What is the
minimiser?
Question 7. (Linear regression)
Let
X
and
Y
be two random variables. We aim to approximate
Y
by a
linear function of
X
. Find
a
and
b
which minimise the square error function
e
(
a, b
) =
E
[(
Y
−
aX
−
b
)
2
]
.

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