SOCIETY OF ACTUARIES
EXAM P PROBABILITY
EXAM P SAMPLE QUESTIONS
Copyright 2015 by the Society of Actuaries
Some of the questions in this study note are taken from past examinations.
Some of the questions have been reformatted from previous versions of thi
Chapter 6
Strong Typicality
Raymond W. Yeung 2012
Department of Information Engineering
The Chinese University of Hong Kong
6.1 Strong AEP
Setup
cfw_X k , k
1, Xk i.i.d. p(x).
X denotes generic r.v. with entropy H (X ) < .
|X | <
6.1 Strong AEP
Denit
Chapter 9
The Blahut-Arimoto
Algorithms
Raymond W. Yeung 2012
Department of Information Engineering
The Chinese University of Hong Kong
Single-Letter Characterization
For a DMC p(y |x), the capacity
C = max I (X ; Y ),
r (x)
where r(x) is the input dist
Chapter 3
The I-Measure
Raymond W. Yeung 2010 Department of Information Engineering The Chinese University of Hong Kong
An Example
H ( X1 ,X 2) H ( X1 X 2) H ( X 2 X1)
H ( X1) I ( X1; X 2)
H (X 2)
Substitution of Symbols
H/I , ; | is some signed measure
Chapter 4
Zero-Error Data Compression
Raymond W. Yeung 2012
Department of Information Engineering
The Chinese University of Hong Kong
4.1 The Entropy Bound
Denition 4.1 A D-ary source code C for a source random variable X is a
mapping from X to D , the s
Chapter 5
Weak Typicality
Raymond W. Yeung 2012
Department of Information Engineering
The Chinese University of Hong Kong
The Notion of Typical Sequences
Toss a fair coin n times.
If the outcome is head approximately half of the time, the sequence of
o
Chapter 1
The Science of Information
Raymond W. Yeung 2010 Department of Information Engineering The Chinese University of Hong Kong
Information Theory
Founded by Claude E. Shannon (1916-2002) The Mathematical Theory of Communication, 1948 Study fundame
CSCI 2100 Tutorial: Priority Queue
Cheng Sheng
([email protected])
Cheng Sheng([email protected])
CSCI 2100 Tutorial: Priority Queue
Review: Priority Queue
Denition
A priority queue maintains a set of numbers, supports:
Insertion (a new number); and
Delete-min.
Example
CSCI 2100 Tutorial: Hashing
Cheng Sheng
([email protected])
Cheng Sheng([email protected])
CSCI 2100 Tutorial: Hashing
Review: Hash Table as a Black Box
What hash table can do
Maintain a set S of integers:
In O (N ) space; and
Test whether q S in O (1) time.
Remark: ac
CSCI 2100 Tutorial: Counting Sort and Linked List
Cheng Sheng
([email protected])
Cheng Sheng([email protected])
CSCI 2100 Tutorial: Counting Sort and Linked List
Review: Counting Sort
Problem
Input: A[1.n] cfw_1, 2, . . . , k .
Output: A , sorted A.
Idea
For each 1 i
CSCI 2100 Tutorial: Complexities
Cheng Sheng
([email protected])
Download slides: sites.google.com/site/csci2100sum11/
Cheng Sheng([email protected])
CSCI 2100 Tutorial: Complexities
Review: Big-O Notations
Pseudo Code
1. for i = 1 to n do
2. for j = 1 to n do
3.
do so
CSCI 2100 Data Structures, Summer 2011
Midterm Examination
Note: Please write all your solutions in the answer book. You can use directly the results
already discussed in the lectures. For example, if you need to perform binary search on an array
of size
CSC2100c Data Structures, Spring 2011
Midterm Examination
Note: Please write all your solutions in the answer book. You can use directly the results already
discussed in the lectures. For example, if you need to perform binary search on an array of size n
Problem
DFS
Topological sort
Lecture 14:
Topological Sort and Depth-rst Search
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
April 8, 2011
Summary
Problem
DFS
Topological sort
In this lecture, we will discuss a problem called topological sort.
Our
Problem
Breadth-rst search
Analysis
Lecture 13: Breadth-rst Search
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
April 3, 2011
Summary
Problem
Breadth-rst search
Analysis
In this lecture, we will discuss breadth-rst search. This is a very
fundamen
Basic concepts
Storage of a graph
Lecture 12: Graph Basics
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
June 28, 2011
Summary
Basic concepts
Storage of a graph
Summary
The next few lectures will be devoted to graphs. Since graphs can
be used to m
Deletion algorithm
Analysis
Lecture 11: AVL-tree 3
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
June 23, 2011
Summary
Deletion algorithm
Analysis
Summary
We have learned how to perform insertions in a binary search tree.
In this lecture, we will
Insertion algorithm
Analysis
Lecture 10: AVL-tree 2
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
March 24, 2011
Summary
Insertion algorithm
Analysis
Summary
In the last lecture, we have learned the static version of the
AVL-tree. In this lecture,
Preliminaries
AVL-tree
Searching the AVL-tree
Lecture 9: AVL-tree 1
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
March 17, 2011
Analysis
Summary
Preliminaries
AVL-tree
Searching the AVL-tree
Analysis
Summary
In this and the next lectures, we will
Problem
Binary tree
Priority queue
Time analysis
Lecture 8: Priority Queue
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
March 12, 2011
Summary
Problem
Binary tree
Priority queue
Time analysis
In this lecture, we will learn a structure called the
The dictionary problem re-visited
Preliminaries
Structure and algorithms
Lecture 7: Hashing
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
February 27, 2011
Analysis of hashing
Summary
The dictionary problem re-visited
Preliminaries
Structure and a
Problem
Structure
Algorithms
Lecture 6: Linked List
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
February 20, 2011
Summary
Problem
Structure
1
Problem
2
Structure
3
Algorithms
Scan
Insert
Delete
Algorithms
Summary
Problem
Structure
Algorithms
In
Problem denition
Counting sort
Lecture 5: Sorting 4
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
February 13, 2011
Summary
Problem denition
Counting sort
Summary
In the last lecture, we showed that all algorithms in the comparison
class have a ti
Comparison class
General behavior of comparison-based sorting
Lower bound of sorting
Lecture 4: Sorting 3
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
January 28, 2011
Summary
Comparison class
General behavior of comparison-based sorting
Lower bo
Quick sort
Analysis of quick sort
Lecture 3: Sorting 2
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
January 20, 2011
Summary
Quick sort
Analysis of quick sort
Summary
This lecture will continue our discussion on the sorting problem.
We will study
Problem
Bubble sort
Merge sort
Lecture 2: Sorting
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
January 17, 2011
Summary
Problem
Bubble sort
Outline
1
Problem
2
Bubble sort
Idea
Algorithm description
Worst-case time complexity
3
Merge sort
Idea
Al
The dictionary problem
Running time of algorithms
Asymptotic complexities
Lecture 1: Complexities
CSC2100 Data Structure
Yufei Tao
CSE department, CUHK
January 17, 2011
Summary
The dictionary problem
Running time of algorithms
Outline
1
The dictionary pro
CSCI 2100 Data Structures, Summer 2011
Final Examination
Note: Please write all your solutions in the answer book. You can use directly the results
already discussed in the lectures. For example, if you need to perform binary search on an array
of size n,