Probability and Statistics I
Week 10
1
1
Chapter 5
Distributions of Functions of random variables
Section 5.1 Functions of one random variable
Question : Let X be a RV of either discrete or continuous type. Consider a function
of X , say, Y u ( X ). Then
Probability and Statistics I
Week 11
1
1
Theorem 5.4-2
Theorem 3.3-2
Proof of theorem 3.3-2
The next two corollaries combine and extend the results of Theorems 3.3-2 and
5.4-2 and give one interpretation of degrees of freedom.
Corollary
Corollary
Chapter
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Probability and Statistics I
Week 5
1
1
Chapter 3
Section 3.1
Continuous distribution
RV of the continuous type
Recall that a RV X: SX(S) R is called a discrete RV if X(S)
is finite or countably infinite.
But RVs with a continuous range of possible values
Probability and Statistics I
Week 9
1
1
Chapter 4
Bivariate Distributions ()
Section 4.3 CONDITIONAL DISTRIBUTIONS
Motivation
Let X and Y have the joint probability mass function f (x, y) : D [0,1].
The marginal pmf of X and Y are
: [0,1] and : 0,1 .
=
Probability and Statistics I
Week 3
1
1
Chapter 2.1
Discrete Distribution
Section 2.1
Random variable of the discrete type
Outcomes of experiment
Numerical
e.g. Rolling a 6-sided die
Not numerical
e.g. Flipping a coin
For the latter case, we can define a
Probability and Statistics I
Week 8
1
1
Chapter 4
Bivariate Distributions ()
Section 4.1 Bivariate Distributions with the discrete type
Motivation
very often, the outcome of a random experiment is a tuple of several things of
interests:
Observe female c
Probability and Statistics I
Week 9
1
1
Chapter 4
Bivariate Distributions ()
Section 4.3 CONDITIONAL DISTRIBUTIONS
Motivation
Let X and Y have the joint probability mass function f (x, y) : D [0,1].
The marginal pmf of X and Y are
: [0,1] and : 0,1 .
=
Probability and Statistics I
Week 6
1
1
Chapter 3
Continuous distribution
Section 3.2
exponential, gamma, chi-Square
Distributions
Definition 3.2-5 [ chi-square distribution]
Let X have a Gamma distribution with = 2, = , r is a integer.
The pdf of X is =
Probability and Statistics I
Week 12
1
1
Chapter 5
Distributions of Functions of random variables
Section 5.7 Approximations for discrete distributions
Motivation: CLT applies to discrete distributions as well. In this
section, we illustrate how the norm
STAT2001 Assignment 4
Do all 6 questions. Show your steps clearly.
Deadline for this assignment is 18th Nov. 5:00p.m. You can submit to the assignment locker
(next to LSB 125) or to your Tutors.
Q1. The random variables X and Y have the joint probability
STA3001 Foundation of Financial and Managerial Statistics
Assignment 2 (for Chapter 3)
Deadline: 3rd March 2017 5:00p.m.
1. Assume the following information about a Treasury zero-coupon yield curve
today:
Maturity (years)
Zero rates (%) Maturity (years) Z
Solution to assignment 5
1
2
1. X ~ N( , ) , f X ( x)
1
2
e
(log y ) 2
E (Y ) Ee M x (1) e
E (Y ) Ee
2x
2 2
, x
1
1
| |
e
y
2 y
2 2
x
2
( x )2
d (log y ) 1
y 0
dy
y
X=logY
gY ( y)
2
e
2
2
4 2
2
e 2 2
2
2. X~U(02)
1/2 if 0 x 2
f x ( x ) cfw_
0 ,
oth
Monty Hall Problem:
Game show host offers you 3 choices.
2 doors behind which hide a goat; the third a car.
Suppose you choose door 1, he then opens door 3. You are given the chance to switch
your choice to door 2.
What is your choice?
The game show host problem
Game show host offers you 3 choices.
2 doors behind which hide a goat; the third a car.
Suppose you choose door 1, he then opens door 3.
You are given the chance to switch your choice to
door 2.
What is your choice?
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Chapter 4: Bivariate Distributions
STAT2001
2016 Term I
Outline
1. Two Discrete Random Variables
2. Two Continuous Random Variables
3. Independence
4. Expectation
5. Covariance and Correlation
(Textbook chapters: 4.1 - 4.4 of 9th edition or 4.1 - 4.3 of 8
Chapter 2: Discrete Distributions
STAT2001
2016 Term I
Outline
1. Discrete random variables
2. Mathematical expectation
3. Binomial distribution and Hypergeometric distribution
4. Moment generating function
5. Poisson distribution
(Textbook chapters: 2.1
STAT2001 Assignment 2
Do all 7 questions. Show your steps clearly.
Deadline for this assignment is 21st Oct. 5:00p.m. You can submit to the assignment locker (next
to LSB 125) or to your Tutors.
1. If the cumulative distribution function of X is given by
1st supplementary note for ch.1
1. Proof of the inclusion-exclusion formula (p.17):
P(C1 .Cn ) p1 p2 p3 . ( 1) n1 pn
n
Remark : p1 P (Ci ), p2
i 1
P (C
i
C j ), p3
1i j n
P (C
i
C j Cl ),.
1i j l n
Use Mathematical Induction (see remark):
A, The for
STAT2001 Assignment 3
Do all 6 questions. Show your steps clearly.
Deadline for this assignment is 9th Nov. 5:00p.m. You can submit to the assignment locker
(next to LSB 125) or to your Tutors. You can also submit electronically through Blackboard.
1. A f
Chapter 1: Fundamental Concepts
STAT2001
2016 Term I
Outline
1. Random experiments and Sample Space
2. Probability
3. Methods of enumeration (Counting)
4. Conditional probability
5. Independent events
6. Bayess theorem
Textbook chapters 1.1 - 1.5 (9th edi
Chapter 3: Continuous Distributions
STAT2001
2016 Term I
Outline
1. Continuous random variables
2. The Uniform distribution and Exponential distribution
3. The Gamma distribution and Chi-square distribution
4. The Normal distribution
(Textbook chapters: 3
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Statistics and Probability |
Confidence Intervals
Sect. 7.1-7.2
Week 12 Given a random sample X1,X2,. .an from a normal distribution N(n,cr2),
we can find a number zap from Table V in Appendix B such that
f- M
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cfw_IA/E _
F
Statistics and Probability l
Conditional Distributions and Joint Distributions for
Continuous Random Variables
Sect. 4.3 4.5, 5.3
Week 6 4.3 CONDITIONAL DISTRIBUTIONS
Definition 4.3-I
The condinnal pmbability mass funetinn of X , given that Y = y, is defi