PROBABILITY COURSE NOTES
STAT/MATH 394/395
Nathaniel Derby
nderby@stat.washington.edu
Summer 2009
University of Washington
Table of Contents
Notation
v
1 Algebra of Sets
1
1.1
Elementary Set Theory
.
1
1.2
Set Functions . . . . . . . . . . . . . . . . . .
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EE 7660
Homework 6
Due May 6, 11
1) Consider a continuous time Markov process cfw_Yt with state space E = cfw_0, 1, 2, .. Denote
the embedded Markov chain (of the states visited by cfw_Yt ) as cfw_Xn . This is the same
notation we have used in class. D
1
EE 7660
Problem Set #5
Due: April 4, 2011
1) A spider hunting a y moves between locations 1 and 2 according to a Markov chain with
transition matrix
.7 .3
,
P1 =
.3 .7
starting in location 1. The y, unaware of the spider, starts in location 2 and moves
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EE 7660
Problem Set #4
Due: March 18, 2011
1) Prove that if the number of states in a Markov chain is M , and if state j can be reached
from state i, then it can be reached in M steps or less.
2) Numbers are drawn consecutively (with replacement) from a
Math 362K Probability
Fall 2007
Instructor: Geir Helleloid
Daily Homework 7 Solutions
1. (Chapter 4, Problem 1) Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win $2 for each black ball selected
1
EE 7660
Problem Set #3
Due Feb 28, 11
1) N = cfw_Nt ; t 0 is a Poisson process with rate .
a) Find E [Nt Nt+s ] for t > 0 and s > 0.
b) Let 0 t1 < t2 < . < tn and let s > 0. Find
P (Ntn +s = k |Nt1 , Nt2 , ., Ntn )
2) Consider a counting process in whic
1
EE 7660
Problem Set #3
Due: March 6, 2013
1) Let cfw_N (t), t 0 be a Poisson process with rate . Let Tn denote the time of the nth
arrival. Find
a) E [T4 ] and E [T4 |N (1) = 2].
b) E [N (4) N (2)|N (1) = 3].
2) cfw_Nt ; t 0 and cfw_Mt ; t 0 are two ind
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EE 7660
Problem Set #2
Due Feb 13, 13
1) N = cfw_Nt ; t 0 is a Poisson process with rate .
a) Find E [Nt Nt+s ] for t > 0 and s > 0.
b) Let 0 t1 < t2 < . < tn and let s > 0. Find
P (Ntn +s = k |Nt1 , Nt2 , ., Ntn )
2) Passengers arrive at a train statio
1
EE 7660
Problem Set #2
Due Feb. 18, 2011
1) A biased coin is tossed repeatedly. Each time there is a probability p of head turning up.
Let Pn be the probability that an even number of heads has occurred after n tosses (zero
is an even number and therefo
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EE 7660
Problem Set #1
Due Feb 4, 11
1) A coin has probability p of showing up heads when tossed. It is successively tossed until
the rth head appears. Let X represent the number of tosses.
a) Find the sample space of X .
b) Show that for n r,
P ( X = n