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
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
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
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 - 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 - 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 #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 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: 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 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
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 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
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
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 Spring 2009 Solutions to Assignment 1
By Juan Manuel Martinez e-mail:j.man.martinez@gmail.com 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
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
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
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: 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
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
Assignment #1 (Due on June 5th in the class)
1. Toss a coin repeatedly and independently. The probability of getting head on each toss
is p with 0 < p < 1. We say a change-over occurs on the ith toss if the outcomes in the
ith and (i 1)th tosses are diere
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
622:
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STAT333 Fall 2014 Tutorial #9
1. Potential customers arrive at a fullservice, one-pump gas station at a Poisson rate of 20 cars
per hour. However, customers will only enter the station for gas if there are no more than
two cars (including the one currentl
Waiting time random variable r.v. and Sum-Product Lemma
Let E be an event and TE be the waiting time for the 1st E. Note that in general, the
possible range of TE is
R = cfw_1, 2, 3, . . . cfw_,
where cfw_ means we can not observe E.
Consider a specific c