MATH3603
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Probability Theory
Assignment 2
Due Date: 22 Oct. 2015 (7:30 pm).
There are 5 exercises. The full mark is 80. The weight of each exercise (full mark) is
shown at the right bottom of each exercise.
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2603 Probability Theory
Solution to Assignment 2 (Updated Version)
Problem 1 (1) By Hint
n=1 nP (x = n)
n
n=1
i=1 P (x = n)
i=1
n=i P (x = n)
P (X i)
i=1
i=0 P (X > i)
(2) E [x] =
=
=
=
=
For the c
Denition 1. For any two random variables X and Y , the conditional expectation
of X given Y is dened to be a function of Y such that for any measurable function
h : R R,
E[Xh(Y )] = E[E[X|Y ]h(Y )].
Proposition 20. Let X and Y be two random variables. Fo
MATH3603/2
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Probability Theory
Exercise 2
1. Let X be a geometric random variable with parameter p.
(a) Compute the moment generating function of X.
(b) Compute the mean and variance of X through .
Solution
MATH3603/1
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Probability Theory
Exercise 1
1. Let E, F , G be three events. Find expressions for the events that of E, F , G
(a) only F occurs,
(b) both E and F but not G occur,
(c) at least one event occurs
Proposition 15 ( Markovs Inequality). If X is a random variable taking only
non-negative values, then for any a > 0, we have
P(X a)
E[X]
.
a
Corollary 3 ( Chebyshevs Inequality). If X is a random variable with nite
expectation and variance 2 , then for
Two events A and B are independent if
P(A B) = P(A)P(B).
A (real-valued) random variable on the probability space (, F, P) is a map
X : R such that
cfw_X b F, b R.
For a random variable X dened on the probability space (, F, P), the cumulative
distribu
Sample Space = the set of all possible outcomes, often denotes by S
Event = collection of one or more of the outcomes of an experiment
For any event E of a sample space S, the complement of E, denoted by E c , is a
new event which consists of all outco
MATH3603
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Probability Theory
Assignment 1
Due Date: 5 Oct. 2015 (6:30 pm).
There are 10 exercises. The full mark is 80. The weight of each exercise (full mark) is
shown at the right bottom of each exercise.
MATHEMATICS 2603
Midterm, October 21, 2009
Show all your work. Use back of page if necessary. You can use your own BLANK scratch paper.
Name:
ID:
1. [10 points] Let X be exponential with mean 1/. Find E [X |X > 1].
Solution: This is a starred problem in t
MATHEMATICS 2603
Midterm, November 27, 2009
Show all your work. Use back of page if necessary. You can use your own BLANK scratch paper.
Name:
ID:
(d)
1. [20 points] State i of a Markov chain has period d. Does that mean Pii > 0? Prove
it or disprove it.
1. Suppose that people arrive at a bus stop in accordance with a Poisson process with
rate . The bus departs at time t. Let X denote the total amount of waiting time of the
person who rst arrives at the bus stop. Let Y denote the total amount of waiting t
1. Read pages 192, 193 of the textbook.
2. Read the remark on page 200 of the textbook.
3. Read Proposition 4.3.
4. [15 points] A stochastic process cfw_Xn , n = 0, 1, 2, with state space S = cfw_0, 1, 2, , l
is called a Markov chain if for any i, j, i0
1. If X is a nonnegative integer valued random variable, show that
E [X ] =
P cfw_X n =
n=1
P cfw_X > n.
n=0
Hint: Dene the sequence of random variables In , n 1, by
In =
1,
0,
if n X
if n > X
Now express X in terms of the In . Can you guess the similar r
Notation: You can either use A (as in class) or Ac (as in textbook) to
denote the complement of set A. For two events E and F , E F (as in class)
or EF (as in textbook) can both be used to denote the intersection.
0. Review the note rst. Then read the tex
1. Suppose that people arrive at a bus stop in accordance with a Poisson process with rate
. The bus departs at time t. Let X denote the total amount of waiting time of the person
who rst arrives at the bus stop. Let Y denote the total amount of waiting t
1. Read pages 192, 193 of the textbook.
2. Read the remark on page 200 of the textbook.
3. Read Proposition 4.3.
4. [15 points] A stochastic process cfw_Xn , n = 0, 1, 2, with state space S = cfw_0, 1, 2, , l
is called a Markov chain if for any i, j, i0
MATH2603 Probability Theory
Solution to Assignment 1
Solution to Problem 1:
Proof. Note that we already prove for ANY two events E1 , E2 ,
P (E1 E2 ) = P (E1 ) + P (E2 ) P (E1 E2 ).
(1)
By induction, assume that we already have for ANY n-1 events E1 , E2
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2603 Probability Theory
Solution to Assignment 3
Problem 26 You have two opponents with whom you alternate play. Whenever you play A, you win with probability pA ; whenever you play B ,
you win wit