Tutorial 5
1
1.1
Review of Chapter 3
Introduction
[Def ] A discrete-time Markov chain is a stochastic process cfw_Xn , n = 0, 1, . . . with a discrete state
space such that S such that
P (Xn+1 = j |Xn
Tutorial 7
1
Review of Chapter 3 (III)
1.1
Limiting probabilities
(n)
[Def ] Let d(i) denote the greatest common divisor of all positive integers n for which pii > 0. The
integer d(i) is called the pe
T4 Solution
1. (a) Suppose the location of the ant just after nth stage is (Xn , Yn ), where Xn and Yn represent
the displacements in West-East direction and North-South direction, respectively. Then,
Tutorial 6
1
Review of Chapter 3 (II)
1.1
Calculation of P n
When |z | < 1,
(I zP )1 = I + zP + z 2 P 2 + . . .
Hence, P n = coecient of z n in expansion of (I zP )1 .
1.2
Classication of states
(n)
T2 solution
1. M follows binomial distribution with parameters (8, 1 ).
2
1
Hence, E (M ) = (8)( 2 ) = 4
1
When M = m, N follows binomial distribution with parameters (m, 2 ).
1
When N = n, Z follows
Tutorial 4
1
Review of Chapter 2
1.1
Random walks
The stochastic process X = cfw_X0 , X1 , X2 , . . . is a random walk in one dimension if
X0 = c where c is a constant.
For n 1, Xn = Xn1 + Zn where
T1 solution
1. (a)
P (X < x, y1 < Y < y2 , Z > z )
=P (X < x)P (y1 < Y < y2 )P (Z > z ) (Because X, Y and Z are independent)
( x
) ( y2
) (
)
=
et dt
et dt
et dt
0
y1
=(1 e
x
)(e
y1
z
y2
e
)e
z
(b) L
Tutorial 2
1
Review of Chapter 1 (II)
1.1
Conditional expectations (II)
[Def ] Suppose X is a random variable and Y is a discrete random variable. Dene a random variable
E (X |Y ) (conditional expecta
T7 solution
1. (a) This is a nite irreducible Markov chain with period 1.
Note that p11 = 0.6 > 0, that means starting from state 1, the chain can go back to state
1 in 1, 2, . . . steps. The period o
T3 solution
1. Note that X is a nonnegative random variable.
(1 F (x)dx
x
=
dx +
e dx
0
[
]
x
= + e
E (X ) =
0
=+
2. Method 1: Note that X is a nonnegative random variable.
E (X ) =
0
1
(1 F (x)dx
(
Tutorial 3
1
Review of Chapter 1 (III)
1.1
Random variables which are neither discrete nor continuous
[Thm1.11] For any nonnegative random variable X , if FX (x) denotes the distribution function of
X
Tutorial 10
1
Review of Chapter 5
1.1
Brownian motion
1
[Lemma] The moment generating function (with t being the argument) of N (, 2 ) is et+ 2
2 t2
.
[Def ] A stochastic process cfw_Xt , t 0 is a Br
STAT2303 & STAT3603
Assignment 2
(Due at 17:00, 10 Oct. 2013)
Q1
Let q 1 p and S be the random sum X1 X2 XN. Then
G X1 (z) p z q.
1A
GS(z) GN(G X1 (z) exp(G X1 (z) 1) exp(p z q 1) ep(z 1),
1M
which is
STAT2303 & STAT3603
Assignment 4
(Due at 17:00, 28 Nov. 2013)
1. Assume that X1 and X2 are random variables following Exp(1) and Exp(2) respectively.
Suppose Y is a nonnegative continuous random varia
T7 Solution
4. Dierent values of k have dierent P .
(a) When k =1,
0
P=
1
(
0
1
)
1/2 1/2
.
1/2 1/2
(b) When k = 2n + 1 where n is a positive integer,
0
0
1/2
1/4
1
0
2
.
.
P= .
.
.
.
k 2 0
k 1 0
k
1
STAT2303 & STAT3603
Assignment 4
(Due at 17:00, 28 Nov. 2013)
Suppose Y has a density f(y). For u 0 and v 0,
Q1
P(X1 Y u, X2 Y v)
0
0
f(y)dy
y u
1exp(1x1)dx1
y v
2exp(2x2) dx2
f(y)exp(1y 1u 2y 2v)dy
STAT2303 Probability Modelling
Class Test (1st semester, 2012~13)
Q1 Q1 of Assignment 2 this year.
Q2 Two gamblers, A and B, initially have capital $k and $(a k) respectively, where k and a
k are pos
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCES
STAT 2303 Probability Modeling (2012-2013)
Example Class 1
Q1. Supp ose that the numb er of road accidents p er week is Pois
STAT2303 & 3603 PROBABILITY MODELLING
Final Exam
What to be examined: all chapters except options in Section 5.2
How many questions: 7
Any results in lecture notes or assignments can be quoted without
STAT2303 & STAT3603
Assignment 3
(Due at 17:00, 11 Nov. 2013)
Q1
(a) Since p01, p10, p12, p21, p23, p32 are all greater than 0, we know that 0 1 2 3. Therefore
there is only one class, which is closed
Tutorial 9
1
Review of Chapter 4
1.1
Denition of Poisson processes
[Def ] A stochastic process cfw_Nt , t 0 is a counting process if Nt represents the total number of
events that have occurred up to t
STAT2303 & STAT3603
Assignment 1
(Due at 17:00, 26 Sept. 2013)
Q1
[Solution 1]
Let X 1A, Y 1B. Let F(x, y) be the joint distribution function of X and Y, and FX(x)
and FY(y) be the marginal distributi
Tutorial 8
1
Review of Chapter 3 (IV)
1.1
Absorbing states
Assume that S = cfw_1, 2, . . . , u, u + 1, . . . , u + v and states 1, 2, . . . , u are absorbing. Then
)
(
Iuu 0uv
P=
Rvu Qvv
Let SA = cfw
T8 Solution
1. (a) Dene 5 states where state i represents that there are i red balls in the urn. The state
space S = cfw_0, 1, 2, 3, 4.
Initial distribution: 0 = (0, 0, 1, 0, 0)
The transition probabi
STAT2303 & STAT3603 Probability Modelling
Class Test (28 Oct 2013)
Q1
h11 P(last shot is by player 1 | the first shot is by player 1 and does not hit the target)
P(the first shot is by player 1 and d
STAT2303 & STAT3603
Assignment 1
(Due at 17:00, 26 Sept. 2013)
1. Suppose there are two events A and B. Prove that
A and B are independent 1A and 1B are independent.
2. Manuscripts are sent to a typin