MATH3075/3975
Financial Derivatives
School of Mathematics and Statistics
University of Sydney
Semester 2, 2021
Tutorial sheet 1
Background: Chapter 6 – Probability Review.
Exercise 1 Assume that the joint probability distribution of the two-dimensional
random variable (X, Y ), that is, the set of probabilities
P(X = i, Y = j) = pi,j for i, j = 1, 2, 3,
is given by:
p1,1 = 1/9, p1,2 = 1/9, p1,3 = 0,
p2,1 = 1/3, p2,2 = 0, p2,3 = 1/6,
p3,1 = 1/9, p3,2 = 1/18, p3,3 = 1/9.
(a) Compute EP(X|Y ), that is, EP(X|Y = j) for j = 1, 2, 3.
(b) Show that the equality EP(X) = EP[EP(X|Y )] holds.
(c) Check if the random variables X and Y are independent.
Exercise 2 The joint probability density function f(X,Y ) of random variables
X and Y is given by
f(X,Y )(x, y) =
1
y
e−x/ye−y, ∀ (x, y) ∈ R2+,
and f(X,Y )(x, y) = 0 otherwise.
(a) Check that f(X,Y ) is a two-dimensional probability density function.
(b) Show that EP(X|Y = y) = y for all y ∈ R+.
Exercise 3 Let X be a random variable uniformly distributed over (0, 1).
Compute the conditional expectation EP(X|X < 1/2).
Exercise 4 Let X be an exponentially distributed random variable with
parameter λ > 0, that is, with the probability density function fX(x) =
1
λ
e−
x
λ
for all x > 0. Compute the conditional expectation EP(X|X > 1).
Exercise 5 We assume that P(X = ±1) = 1/4, P(X = ±2) = 1/4 and we
set Y = X2. Check whether the random variables X and Y are correlated
and/or dependent.
Exercise 6 (MATH3975) Let U and V have the same probability distribu-
tion and let X = U + V and Y = U − V . Examine the correlation and
independence of the random variables X and Y (provide relevant examples).
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