STA3007
Applied probability (08-09) Chapter 6
Solution to Assignment 11
Problem 4.1
If there is single repairman. As N = M = 5, R = 1, = 2, = 1, we have k = = 2, k = 0, 1, 2, 3, 4, 5 = 0, k = k = k, k = 0, 1, 2, 3, 4, 5, 4 2 4 0 = 1, 1 = 2, 2 = 2, 3 = , 4
STA3007
Applied probability (08-09) Chapter 6
Solution to Assignment 10
Problem 2.1
Let Wi be the waiting time of the ith death, and W0 = 0, Si be the sojourn time in the ith state, we have Pr(X (T ) = 0) = Pr(T > WN )
N
=
i=1 N
Pr(T > Wi |T > Wi1 ) Pr(T
STA3007
Applied probability (08-09) Chapter 5
Solution to Assignment 9
Problem 6.2
The key observation is that, if is uniform on [0, 2 ), and Y is independent of and has an arbitrary distribution, then Y + (mod 2 ) is uniform on [0, 2 ). The proof of this
STA3007
Applied probability (08-09) Chapter 5
Solution to Assignment 8
Problem 3.1
If W1 > w1 , W2 > w2 , then there is no event occurring in the interval [0, w1 ], and there may 0 or 1 event occurring in the interval (w1 , w2 ], which imply X (w1 ) = 0,
STA3007
Applied probability (08-09) Chapter 5
Solution to Assignment 7
Problem 1.1
Let X = 1 + + k and Y = k+1 , then X and Y are independently distributed as Gamma distribution with pdf fk (x) = k ex xk1 (k 1)!
and exponential distribution with parameter
STA3007
Applied probability (08-09) Chapter 4
Solution to Assignment 5
Problem 1.2
Let Xn be the number of balls in transition matrix is as follows 0 1 5 0 P = 0 0 0 urn A, the state space is cfw_0, 1, 2, 3, 4, 5. The 100 040 5 2 03 5 5 030 5 4 005 000 00
STA3007
Applied probability (08-09) Chapter 3
Solution to Assignment 4
Problem 4.1
Method I Let 1, 2, 3, 4, 5, 6, 7, 8 denote the dierent patterns of three tosses, HHH, HHT, HTH, HTT, THH, THT, TTH, TTT, respectively. Then, we can get the following transi
STA3007
Applied probability (08-09) Chapter 3
Solution to Assignment 3
Problem 1.1
The state space is cfw_0, 1, 2, 3, 4, 5.It is obvious that P00 = P55 = 1. If there are k (k = 1, 2, 3, 4) people who are diseased, the number of pairs of individuals in whi
STA3007
Applied probability (08-09) Chapter 2
Solution to Assignment 2
Problem 5.1
Let X, Y, Z denote Xn+2 , Xn+1 , (X0 , , Xn )T , then, according to the given formula E[E(X |Y, Z )|Z ] = E[X |Z ], we have E[Xn+2 |X0 , , Xn ] = E[E(Xn+2 |X0 , , Xn+1 )|X0
STA3007
Applied probability (08-09) Chapter 1
Solution to Assignment 1
Problem 2.3
Let Sr = 1 + + r , where k is the number of additional samples needed to observe the k th distinct element compared to observing the (k 1)th distinct element. Assume that t
STA 3007
Assignment No. 11
No need to hand them in,
All problems are from the book by Taylor & Karlin, 3rd ED Exercises: Chapter VI: Chapter VII: Problems: Chapter VI: Chapter VII: 4.2, 4.4 1.1, 1.3, 1.4
4.1, 4.2, 4.3, 4.7 1.1, 1.2, 1.3
STA 3007
Assignment No. 10
Due Wednesday, Nov. 19, 2008,
All problems are from the book by Taylor & Karlin, 3rd ED Exercises (no need to hand in): Chapter VI: 2.2, 2.4, 3.1, 3.2
Problems (have to hand in): Chapter VI: 2.1, 2.2, 2.4, 2.6, 3.1, 3.3
STA 3007
Assignment No. 9
Due Wednesday, Nov. 12, 2008,
All problems are from the book by Taylor & Karlin, 3rd ED Exercises (no need to hand in): Chapter V: 6.1, 6.2 Chapter VI: 1.1, 1.3, 1.6
Problems (have to hand in): Chapter V: 6.2, 6.3, 6.6 Chapter VI
STA 3007
Assignment No. 8
Due Wednesday, Nov. 12, 2008,
All problems are from the book by Taylor & Karlin, 3rd ED Exercises (no need to hand in): Chapter V: 3.2, 3.4, 4.1, 4.4, 5.1, 5.3
Problems (have to hand in): Chapter V 3.1, 3.3, 3.6, 4.1, 4.7, 5.1, 5
STA 3007
Assignment No. 7
Due Wednesday, Nov. 5, 2008,
All problems are from the book by Taylor & Karlin, 3rd ED Exercises (no need to hand in): Chapter V: 1.1, 1.3, 1.4, 1.5, 2.1, 2.2, 2.4
Problems (have to hand in): Chapter V 1.1, 1.2, 1.5, 1.9, 2.2, 2.
STA 3007
Assignment No. 6
Due Wednesday, Oct 30, 2008,
All problems are from the book by Taylor & Karlin, 3rd ED Exercises (no need to hand in): Chapter IV: 3.2, 3.3, 3.4, 4.1, 4.2, 5.2
Problems (have to hand in): Chapter IV 3.1, 3.2, 4.1, 4.3, 4.4, 4.7,
STA 3007
Assignment No. 5
Due Wednesday, Oct 15, 2008,
All problems are from the book by Taylor & Karlin, 3rd ED Exercises (no need to hand in): Chapter IV: 1.2, 1.4, 1.10, 2.1, 2.2, 2.3, 2.6
Problems (have to hand in): Chapter IV: 1.2, 1.3, 1.4, 1.8, 1,1
STA 3007
Assignment Week 7
Due Wednesday, October 2, 2008,
All problems are from the book by Taylor & Karlin, 3rd ED Exercises (no need to be handed in): Chapter 3 (page 127): 4.2, 4.8, 5.4, 5.6, 5.8, 6.1, 6.2
Problems (have to be handed in): Chapter 3 (p
STA 3007
Assignment Week 3
Due Wednesday, September 24, 2008,
All problems are from the book by Taylor & Karlin, 3rd ED Exercises (no need to be handed in): Chapter 3 (page 99): 1.4, 1.5, 2.5, 2.6, 3.4, 3.5 Problems (have to be handed in): Chapter 3 (page
STA 3007
Assignment Week 2
Due Wednesday, September 17, 2008,
All problems are from the book by Taylor & Karlin, 3rd ED Exercises (no need to be handed in): Chapter 2 (page 93): 5.2, 5.3 Chapter 3 (page 183): 8.1, 8.3 Problems (have to be handed in): Chap
STA 3007
Assignment Week 1
Due Wednesday, September 10, 2008,
All problems are from the book by Taylor & Karlin, 3rd ED Exercises (no need to hand in): Chapter 1: 2.6, 2.9, 2.10, 3.2, 3.4, 4.1, 4.2, 4.5, 4.6, 5.4 Chapter 2: 1.1, 1.4, 1.5 Problems (have to