12/13 Spring
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601 Probability & Statistics I
Assignment 2
Due Date: March 18, 2013
(Hand in your solutions for Questions 2, 13, 19, 21, 25, 34, 45, 52)
1. The probability mass
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601
PROBABILITY AND STATISTICS I
EXAMPLE CLASS 9
Review
Conditional distribution and conditional expectation
For any two events E and F, the conditional probability of E given
Assignment 1, Questions 20.
Twenty people get on an elevator at the ground floor of a building. If each person
is equally likely to choose any of the 10 floors, independently, what is the
probability that the elevator stops at all 10 floors?
Solution:
Den
Example class 10
STAT2601 Probability and Statistics I
Kevin Nailin Li
April 21, 2013
Review: Transformation of Multivariate Distribution I
Let X1 , X2 , . . . , Xn be jointly distributed continuous random variables with
joint probability density function
Example class 9
STAT2601 Probability and Statistics I
Kevin Nailin Li
April 21, 2013
Review: Conditional distribution and conditional
expectation I
For any two events E and F, the conditional probability of E given F is
dened by
P(E |F ) =
P(E \ F )
P(F )
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601 PROBABILITY AND STATISTICS I
EXAMPLE CLASS 10
Review
Transformation of Multivariate Distribution
Let X1 , X2 , . . . , Xn be jointly distributed continuous random variable
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601
PROBABILITY AND STATISTICS I
EXAMPLE CLASS 9
Review
Conditional distribution and conditional expectation
For any two events E and F, the conditional probability of E given
Example class 6
STAT2601A Probability and Statistics I
Kevin Nailin Li
March 21, 2013
Review:Some continuous distributions
Gamma Function and Beta Function
() = 0 e x x 1 dx , > 0
(+
1
Beta(, ) = 0 x 1 (1 x ) 1 dx = ()()
Uniform Distribution
Let X U(a, b)
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601 PROBABILITY AND STATISTICS I
EXAMPLE CLASS 10
Review
Transformation of Multivariate Distribution
Let X1 , X2 , . . . , Xn be jointly distributed continuous random variable
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601 PROBABILITY AND STATISTICS I
EXAMPLE CLASS 8
Review
Covariance
Let X and Y be random variables with mean x and y respectively. The covariance between X and Y, denoted by x
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601A
PROBABILITY AND STATISTICS I
EXAMPLE CLASS 6
Review
Some Common Continuous Distributions
Gamma Function and Beta Function
1
() = 0 ex x1 dx, > 0
Beta(, ) = 0 x1 (1 x)1 dx
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601
PROBABILITY AND STATISTICS I
EXAMPLE CLASS 7
Review
Quantiles of a distribution
The
quantile of a probability distribution function FX of a random variable X is dened to b
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601
PROBABILITY AND STATISTICS I
EXAMPLE CLASS 7
Review
Quantiles of a distribution
The
quantile of a probability distribution function FX of a random variable X is dened to b
Stat 2601
Probability & Statistics I
Spring 2014-2015
Chapter VII Sampling Distributions and Large Sample
Theories
7.1
Population and Sample
Population : a group of individuals about which we wish to make an inference.
We usually do not gather informatio
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601
PROBABILITY AND STATISTICS I
EXAMPLE CLASS 1
Review
Combinatorial Analysis
(a) Multiplication Principle
(b) Selection of r from n distinct objects:
Ordered
Unordered
With
Stat2601
Chapter I
1.1
Probability & Statistics I
Spring 2014-2015
Origin of Probability and Combinatorial
Analysis
Introduction
Presence of uncertainty :
An investor chooses between buying stocks and tying up assets in real estate in
an uncertain econo
Stat2601 Probability & Statistics I
Chapter V
Spring 2014-2015
Special Distributions
In applications of probability, certain families of distributions arise quite frequently
and it is important to have a thorough understanding of these frequently occurrin
14/15 Spring
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601 Probability & Statistics I
Assignment 4
Due Date: May 8, 2015
(Hand in your solutions for Questions 2, 7, 12, 20, 27, 32, 33, 38, 47, 49)
1. Find the lower qu
14/15 Spring
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601 Probability and Statistics I
Assignment 4 Solution
2. The cumulative distribution function of X is given by
2.5200
F x
200
t 3.5
t
2.5
2.5
200 2.5
200
d
14/15
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601 Probability and Statistics I
Assignment 3 Solution
7. (a) The radius of the circle defined by S is 9 3 and hence he area of S is 32 9 .
Since X , Y is uniformly dist
14/15
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601 Probability & Statistics I
Assignment 3
Due Date: April 17, 2015
(Hand in your solutions for Questions 7, 15, 25, 29, 38, 39, 54, 61, 69, 78)
1. Trains headed for de
14/15
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601 Probability and Statistics I
U
Assignment 2 Solution
U
2. (a) E X 3 0.03 4 0.05 13 0.01 7.9
The expected number of years of patent life for a new drug is 7.9 years.
14/15
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601 Probability & Statistics I
Assignment 2
Due Date: March 20, 2015
(Hand in your solutions for Questions 2, 13, 26, 30, 36, 42, 45, 55, 58, 64)
1. The probability mass
Stat2601 Probability & Statistics I
Chapter IV
4.1
Spring 2014-2015
Mathematical Expectation
Expected Value
Example 4.1
Consider the following two games. In each game, three fair dice will be rolled.
Game 1: If all the dice face up with same number, then
Stat2601 Probability & Statistics I
Chapter III
3.1
Random
Variables
Distributions
and
Spring 2014-2015
Probability
Random Variables
Definition
A random variable X : is a numerical valued function defined on a sample
space. In other words, a number X , p
Stat2601
Chapter II
Probability & Statistics I
Spring 2014-2015
Mathematical Theory of Probability
In 1933 a Russian mathematician named Andre Nikolayevich Kolmogorov
published the axiomatic structure of probability theory. A review of the essential
conce
Stat2601 Probability and Statistics I
Chapter VI
6.1
Spring 2014-2015
Transformation of Random Variables
Transformation of Univariate Distribution
In general, functions of random variables are random variables.
Example 6.1
1. If Z ~ N 0,1 , then
X Z ~ N
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601 PROBABILITY AND STATISTICS I
EXAMPLE CLASS 8
Review
Covariance
Let X and Y be random variables with mean x and y respectively. The covariance between X and Y, denoted by x
Example class 4
STAT2601 Probability and Statistics I
Kevin Nailin Li
February 26, 2013
Review
Abbreviated event notation
Usually we abbreviate the description of the event or the set
cfw_ : X () x to simply as cfw_X x so that we can also abbreviate
the
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2601
PROBABILITY AND STATISTICS I
EXAMPLE CLASS 4
Review
Abbreviated event notation
Usually we abbreviate the description of the event or the set cfw_ : X() x to simply as cfw_