Solution of Assignment 1
1. (a) Let event E denote that 1 appears before 6 in the sequence. Conditioning on
the outcome of the rst toss, we have:
6
P (E |1st = i)P (1st = i)
P (E ) =
i=1
=
14
+ P (E ).
66
Note that if the rst outcome is 2, or 3, or 4, or
Assignment #1 (Due on Oct. 11th in the class)
1. Suppose we toss a fair die repeatedly and independently. We get a sequence of numbers.
(a) Find the probability that 1 appears before 6 in the sequence.
(b) Find the probability that 1 or 2 appear before 4
Stat 333
Spring 2009
Assignment #1
The assignment is due Tuesday, May 26 in class. It will be posted in segments, and I will let you know when all segments have been posted and the assignment is complete. All problems are to be turned in. Only a random su
STAT 333 - Spring 2014 - Assignment 2 SOLUTIONS
1. Suppose we toss a fair coin repeatedly and observe a sequence of H or T. Let be the event
T T H H.
(a) Why is a renewal event?
Because it is a pattern with no overlaps. Once the event occurs, it does not
STAT 333 - Spring 2014 - Assignment 1
Due Thursday June 5 at start of class
This assignment may be completed in pairs. Only one person should submit the paper. Both
names and ID numbers should be on it. Both will receive the same grade.
1. Five people are
Stat 333 - Assignment 1 Solutions - Fall 2011
1.
(a) We will dene random variables Ij to indicate the state of computer j , j = 1, 2, 3, 4.
Therefore,
Ij =
, if computer j is isolated
, if computer j is not isolated
1
0
Computer j is isolated iff all of t
STAT 333: Assignment #3
Due: Dec. 1st in Pengfeis oce no later than 4pm.
Note: Please use the cover page for the assignment. No cover page=0%.
1. Consider the Markov chain with the state space S = cfw_0, 1, 2, 3, 4 and the transition matrix
0
0
1
P = 2
3
STAT 333: Assignment 1
Due: October 9th in class, or in Pengfei or Martins oce no later than 5pm on October 10th.
Note: Please use the cover page for the assignment.
1. A collection of n coins is ipped. The outcomes are independent and the ith coin comes
Stat 333
Winter 2009
Assignment #3
The assignment is due Friday, April 3 , in class. Questions 14 posted March 14: These should be done before the term test 1. (a) Letters are drawn randomly with replacement from the set cfw_E, H, R, S, T . (i) Find the e
STAT 333: Assignment #3
Due: Dec. 1st in Pengfeis oce no later than 4pm.
Note: Please use the cover page for the assignment. No cover page=0%.
1. Consider the Markov chain with the state space S = cfw_0, 1, 2, 3, 4 and the transition matrix
0
0
1
P = 2
3
STAT 333 Assignment 1 SOLUTIONS
1. Consider 10 independent coin flips having probability p of landing heads. a. Find the expected number of changeovers. Let Xi = 1 if there is a changeover between trial i and trial i+1, 0 otherwise, i = 1, 2, ., 9. Then X
STAT 333 Assignment 3 SOLUTIONS
1. At all times, a container holds a mixture of N balls, some white and the rest black. At each
step, a coin having probability p, 0 < p < 1, of landing heads is tossed. If it is heads, a ball is
chosen at random from the c
STAT 333 Assignment 1
Due: Thursday, May 29 at the beginning of class 1. Consider 10 independent coin flips having probability p of landing heads. We say a changeover occurs whenever an outcome differs from the one preceding it. For example, if the result
STAT 333 Assignment 1 SOLUTIONS
Due: Thursday, Sep. 30 at the beginning of the class
Chapter 1
1. (#18)Assume that each child who is born is equally likely to be a boy or a girl. If a
family has two children, what is the probability that both are girls gi
STAT 333: Applied Probability
Homework 3
Instructions:
Due Monday, April 6 at 5pm in Drop Box 15, Slots 2 & 4.
Include the names of all collaborators.
1. An auto insurance company receives claims from clients according to a Poisson sprocess
with rate = 50
STAT 333: Assignment #3
(Due: July 30th before 3:00pm in my oce)
1. Consider the Markov chain with the state space S = cfw_0, 1, 2, 3, 4 and the transition matrix
0
0 1/5
1 0
P = 2 0
3 0
4 0
1
1/5
1/3
0
3/5
0
2
1/5
0
1/2
0
1/2
3
0
2/3
0
2/5
0
4
2/5
0
1/2
Question 1. (10pts)
Let X1, X2, X 3 . , . be independent and identically distributed random variables from Binomial distribution
with size 2 and probability 0.5. That is, for any n 2 1,
P(Xn : 0) z 0.25, P(Xn = 1) = 0.5, P(Xn = 2) : 0.25.
Suppose N N Geo(
Solutions for Practice Questions in Chapter 4.
1. (a)
s
5 + 3s
1
s
5 1 + 3s/5
C (s) =
=
Using the alternative geometric series, we have
C (s) =
s
3
1
s
s
=
5 1 + 3s/5
5 n=0
5
n
=
1
3
5
5
n=0
n
sn+1 .
3
Note that the convergence radius is 5 since the commo
Answers for the review problems about the basic concepts of the
probability
Chapter 1
1. Let A and B be two events.
(a) Show that in general, if A and B are disjoint, then they are not necessarily independent.
Solution: if A and B are disjoint, then
AB =
Stat 333
Winter 2009
Assign #1
Selected Partial Solutions
1. A is right and the guard is wrong. That is, if we let B = the guard says B is going free, then P (A executed) = P (A executed|B ) = 1/3. Intuitively this occurs because the guard is only allowed
STAT 333 Assignment 1
Due: Thursday, May 27 at the beginning of class 1. Consider rolling a fair 6-sided die n times. Let X represent the number of faces that have NOT been rolled. a. Find the expected value of X. b. Find the variance of X. c. Describe (i
STAT 333 Spring 2009 Solutions to Assignment 1
By Juan Manuel Martinez e-mail:[email protected] June 3, 2009
Type I Problems
Problem 1
a) Here T has an exponential distribution with parameter = 2. Its cumulative distribution function is therefore g
STAT 333 Assignment 1
Due: Thursday October 6 at the beginning of class
1. Suppose four computers are joined in a network according to the following diagram. (The
double lines indicate two separate connections.) Each connection has probability p of
being
Chapter 7
Continuous markov process
In this chapter, we are still interested in the continuous process
cfw_X(t), t 0.
The following are some concepts related to the continuous process.
1. State space (S): all the possible values of cfw_X(t), t 0. We still
STAT 333 Assignment 2 SOLUTIONS
Due: Thursday, June 26 at the beginning of class 1. Suppose we toss a fair coin repeatedly and observe a sequence of H or T. Let be the event H H T T . a. Why is a renewal event? There is no overlap, so knowing that the eve
STAT 333: Assignment 2
Due: Nov. 7th in class, or in Pengfei or Martins oce no later than 4pm on Nov. 8th.
Note: Please use the cover page for the assignment. No cover page=0%.
1. Let X1 , X2 , X3 , . . . be independently and identically distributed Berno
Assignment #2 (Due on Nov 13th in the class)
Note 1: Please use the cover page for this assignment.
Note 2: The pgf of Geo(p) rv is
ps
.
1(1p)s
You can directly use it without proof.
1. Suppose X NegBin(r, p). Find the probability generating function of X