The Hong Kong University of Science and Technology
PROBA
MATH 2

Spring 2014
HKUST
Math 2421: Probability
Midterm Examination (Spring 2012)
Name:
22th March 2012
19:0021:00
Student I.D.:
Tutorial session:
Directions:
This is a closed book examination and the exam will be
2 hours long.
Write down your FULL name, student ID number
The Hong Kong University of Science and Technology
PROBA
MATH 2

Spring 2014
Bivariate transformation in discrete cases:
Consider two discrete random variables X1 and X2 with their joint pmf pX1 ,X2 . Let
Y1 = g1 (X1 , X2 ) and Y2 = g2 (X1 , X2 ). To nd the joint pmf of Y1 and Y2 , we rst dene the
following sets:
X2 = cfw_(x1 , x2
The Hong Kong University of Science and Technology
PROBA
MATH 2

Spring 2014
Chapter Four
1. Joint and Marginal Probability
Distributions
1
1. Joint Probability
Distributions
In previous chapter, we examined situations with respect to a single random variable.
That is, we restricted ourselves to onedimensional analysis. However,
The Hong Kong University of Science and Technology
PROBA
MATH 2

Spring 2014
(MATH241)[2010](s)midterm~1365^_375.pdf downloaded by zhuangab from http:/petergao.net/ustpastpaper/down.php?course=MATH241&id=0 at 20140206 16:35:58. Academic use within HKUST only.
HKUST
MATH 241 Probability
Midterm Examination
Name:
27th March 2010
S
The Hong Kong University of Science and Technology
PROBA
MATH 2

Spring 2014
(Math241)[2010](s)midterm~1365^_28668.pdf downloaded by zhuangab from http:/petergao.net/ustpastpaper/down.php?course=MATH241&id=1 at 20140206 16:36:04. Academic use within HKUST only.
(Math241)[2010](s)midterm~1365^_28668.pdf downloaded by zhuangab fro
The Hong Kong University of Science and Technology
PROBA
MATH 2

Spring 2014
Chapter Five
1. Covariance, Correlation and
Conditional Expectation
1
1. Expectation of the sum of random
variables
2
1. Expectation of the sum of random
variables
Independence assumption is not required.
3
4
5
6
2. Expectation of a product of random
vari
The Hong Kong University of Science and Technology
PROBA
MATH 2

Spring 2014
Thus,
2
3
4
5
6
Inclass problem (5)
Find the density function
7
Suppose that X and Y are continuous random
variables.
Recall that
and
10
Suppose that X and Y are continuous random
variables.
And they are independent.
Recall that
and
11
Inclass problem (
The Hong Kong University of Science and Technology
PROBA
MATH 2

Spring 2014
lVLmTRéZ (GL1 kei (Bram CSPWS om)
Consider a circle of radius R, and suppose that a point within the circle is randorxiiy chosen
in such a, manner that aii region within the circie of equal area are equally likely to contain
the point. In other words, th
The Hong Kong University of Science and Technology
PROBA
MATH 2

Spring 2014
MATH 2421 Tutorial
Sep. 9th 2013.
SONG Yang (ysongad@ust.hk)
Why is counting important to probability, and why is hard.
Principles of Counting.
Techniques of Permutations and Combinations.
1
MATH241 Tutorial 1
February 10, 2009
by Daniel Zheng
. .
(a) Exa
The Hong Kong University of Science and Technology
PROBA
MATH 2

Spring 2014
MATH241 Tutorial 2
16th, Sept., 2013
MATH 2421 Tutorial
SONG Yang (Ysongad@ust.hk)
February 18, 2009
by Daniel Zheng
. .
(a) Theoretical Exercise 10 (page 19)
Prove the combinatorial identity:
k
n
k
= (n k + 1)
n
k
=n
+
n1
k1
.
Solution. From a group of n