Probability I - Hw 9
Sunhwa
2015
(1) Let cfw_Xt : t 0 be a CMTC on the state space = cfw_1, 2, 3 with
the following transition rates:
12 = 1, 21 = 10
23 = 1, 32 = 5
and all other rates zero.
(a) Find the stationary distribution i
The Generator of the chai

Probability I - Homework 2
Sunhwa
Sep 2015
10) There are two dice on the table. One is a fair die, the other
is xed so that it always comes up showing an even number. You
pick one die at random and roll it, getting an even number. Find the
probability tha

Probability I - Homework 1
Sunhwa
Sep
1) Given three events E, F, G, write formulas for the following
events: only E is true; both E and F but not G; at least two of
the events are true
i Let E, F, G S be events (sets of possible outcomes), then let A be

Probability Theory: Ch 1 Problems
September 6, 2016
Exercise 1 Given three events E, F, G, write formulas for the following events: only E is true;
both E and F but not G; at least two of the events are true.
Exercise 2 Two events E and F ; the probabilit

Ch. 1: Elementary Probability Theory
Christopher King, Northeastern University
August 4, 2016
0.1
Basics: sample space, events, probability law
Probability theory provides the tools to organize our thinking about how to answer
questions involving randomne

Probability Hw 7
Sunhwa
2015
(1) A random walk is dened on the integers cfw_ 0,1,2,3,. . . with
the following transition probabilities
1
1
p00 = , p01 =
2
2
1
pn,n+1 = pn,n1 = ; n = 1, 2, . . .
2
Determine whether the walk is transient or persistent. [Hi

Probability Hw 8
Sunhwa
January 28, 2016
(1) A simplied version of the Google page-rank algorithm assigns
a probability distribution R(1), . . . , R(n) to a collection of n webpages.
The webpages are connected by directed links. For each page i, let
L(i)

Probability I - Homework 5
Sunhwa
January 28, 2016
(1) A transition matrix is doubly stochastic if each column sum is
1. Find the stationary distribution for a doubly stochastic chain with
M states.
Let i 1, 2, . . . , M and, j 1, 2, . . . , M , ij be the

Probability I - Homework 6
Sunhwa
2015
(1) For a Markov chain, suppose that state i is transient, and state
i is accessible from state j (meaning that there is some integer m s.t.
pji (m) > 0). Show that pij (n) 0 as n
State i being transient tells us th

Probability I - Homework 4
Sunhwa
2015
(26) Randomly distribute r balls in n boxes so that the sample
space consists of nr equally likely elements. Let Nn be the number of
empty boxes. Verify that (1 n1 )r is the probability that the i th
box is empty. Su

Probability I - Homework 3
Sunhwa
2015
18) A rat is trapped in a maze with three doors. Door #1 leads
to the exit after 1 minute. Door #2 returns to the maze after three
minutes. Door #3 returns to the maze after ve minutes. Assuming
that the rat is at al