Chapter
1.7 Prove Bonferronis inequality :
P( A B) P(A) + P(B) 1
SOL :
P(A B) = P(A) + P(B) P(A B) 1
P( A B) P(A) + P(B) 1
1.8 Prove that
n
n
) P ( Ai )
P( Ai
i
1
i
1
SOL :
prove that
n
n
P( Ai
i
1
n=2
2
P( Ai
i
1
) P ( Ai )
i
1
2
) = P(A 1
A 2 ) = P(A
Ch2 Homework Solution
2-8 Show that the binomial probabilities sum to 1.
X ~ B (n, p)
n
n
k p k (1 p)nk = ( p + (1 p)n = 1
k = 0
2-11 Consider the binomial distribution with n trails and probability
p of success on each trial. For what value of k is P
SS3657 Lecture 28
Miscellaneous items
Moment approximations
QQ plot
Sample Mean and Variance for iid Normals
1/15
Taylors series of a function g
Taylors series of a function g, about x0 is
1
g(x) = g(x0 )+g (x0 )(xx0 )+ g (x0 )(xx0 )2 +higher order(xx0 )
SS3657 Lecture 20
Methods involved in calculating expectations/moments.
Various examples
1/11
Example 1
X = the number or trials (tosses) till the rst success.
X Geometric( p) with pmf
f (k) = P (X = k) = (1 p)k1 p, k = 1, 2, . . .
where 0 < p < 1.
Calcul
SS3657 Lecture 19
Conditional Expectation
Review Denition of conditional expectation
Properties of conditional expectation
Examples
Textbook coverage: Chap 4.4
1/8
Denition of Conditional Expectation
Denition The conditional expectation of Y given X is de
SS3657: Intermediate Probability
Course outline and course plan (available online in OWL)
Textbook and contents covered in this course
The distinctions between SS3657 and SS2857
The importance of this course
Important dates (see course plan)
Course
SS3657 Lecture 18
Conditional Expectation
Review an example from Lecture 8 and 9
Denition of conditional expectation
Properties of conditional expectation: Theorem A
Examples illustrating the calculation of E(Y |X) and usefulness
of Theorem A.
Textbook co
SS3657 Lecture 17
Moments of a r.v. X or more precisely of the distribution of X.
Denition of moments of random variables
Intuition of moments (mean, variance, skewness and kurtosis)
Tail probability and Chebychevs inequality
Textbook coverage: Chap 4.1-4
SS3657 Lecture 16
Expectation
Review the denition of expectation and various formula for
calculating expectation
What is new about expectation?
Understand there is only one denition for expectation
Understand the existence condition for expectation and ho
SS3657 Lecture 15
Order Statistics
Textbook coverage: Chap 3.7
1/12
Order Statistics and Continuous Random Variables
Consider a random sample X1 , X2 , . . . , Xn with common
continuous distribution (F or f ). Thus Xi , i = 1, 2, . . . , n are
i.i.d. F (o
SS3657 Lecture 1
Review old concepts with new perspective:
Examples in the form of clicker questions
The probability of obtaining a sum of 7 when throwing two
dices
The birthday problem
The concept of probability triplet (, F, P)
The axioms of probability
SS3657 Lecture 13
Transformations
A special case: ratios
Two approaches for Z = Y /X
Textbook coverage: Chap 3.6
1/12
Examples on Ratios Continuous Case
Suppose that X, Y are iid Uniform(-1,1) random variables. Let
Y
Z = X.
In this handout we nd the pdf o
SS3657 Lecture 10
Extension to Continuous Multivariate Distribution
Remarks on independent random variables
Textbook coverage: Chap 3.4
1/9
Continuous Multivariate Distributions
The d dimensional joint cdf for random variables X1 , . . . , Xn is a
functio
SS3657 Lecture 14
Review methods for transformation Y = g(X)
Besides the formulas that have been derived so far, the most
important is the ability to work with
FY (y) or P (Y B) = B f (y)dy = B h(y)dy
Notice that Y can be of d-dimension, so is .
Examples
SS3657 Lecture 12
Transformations
General comments on transformations
A special case: ratios
Textbook coverage: Chap 3.6
1/14
General Comments on Transformations and Distributions
This section studies how to nd the distribution of Y = g(X),
where Y may be
SS3657 Lecture 2
It is all about conditional probability
Examples in the form of clicker questions: Polyas Urn model
The denition of conditional probability
The Law of Total Probability
The Bayes Rule
Textbook coverage: Chapter 1: 1.5
1/8
The denition of
SS3657 Lecture 3
Examples in the form of clicker questions: coin tossing, circuit
example.
The concept of independence
Pairwise independence v.s. (mutually) independence
Physical independence v.s. statistical independence
Mutually exclusive events are act
SS3657 Lecture 29
Review the common mistakes in Test 2
Information for the nal exam
Review concepts learned in this course
Arrangement before the nal
1/9
Q4 of Test 2
1. Understand that you need to get a joint pdf, say, fU,V (u, v) or
f1,4 (u1 , u4 ), whi
SS3657 Lecture 27
Convergence in probability
The Law of Large Numbers
Some applications of LLN
Textbook coverage: Chap 5.2
1/8
Convergence in Probability
Denition (Convergence in Probability) A sequence of r.v.s
Xn , n = 1, 2, 3, . . . is said to converge
SS3657 Lecture 21
Review some of the results w.r.t. joint (X, Y )
Various examples integrating questions from chapters 3 and 4
1/6
Derive the joint pdf under invertible di. transformation
Assume a transformation from
(X, Y ) (U, V ) = g(X, Y ) = (g1 (X, Y
SS3657 Lecture 24
Moment Generating Functions (mgf)
Review and remark on mgf
Recall Property A: If the mgf exists for t in an open interval
containing zero, it uniquely determines the distribution.
Application of Property A in studying sums of independent
SS3657 Lecture 26
The Central Limit Theorem (CLT) (to be proved and
explained)
Specic examples illustrating CLT
Applications of CLT: Prediction interval for sample mean, etc.
Textbook coverage: Chap 5.3
1/13
Example 4 from last lecture
Suppose that Xi , i
SS3657 Lecture 22
Review for midterm test 2
Simulations that enhance understanding of Cov(X, Y )
1/7
Questions on Test 2
1. A denition, a proof and 4 T or F statement : whatever after
test 1
2. conditions are provided
(a) Find E(X|Y )
(b) Find E(X) using
SS3657 Lecture 23
Moment Generating Functions (mgf)
Denition and properties of mgf
MGF is a useful tool for
calculating moments
studying sums of independent r.v.s
studying limiting theorems
Textbook coverage: Chap 4.5
1/11
Denition of mgf
Denition (Moment
SS3657 Lecture 25
Convergence in Distribution
Denition and remarks regarding convergence in distribution
Continuity Theorem
Examples of using Continuity Theorem to characterize the
limit distribution
The Central Limit Theorem
Textbook coverage: Chap 5.3
1
SS3657 Lecture 9
More on conditional distributions
examples on conditional distributions
An examples of mixture distribution
Textbook coverage: Chap 3.5
1/10
Example 1
For bivariate r.v. (X, Y ) in Example 1 of Lecture note 4, where
(X, Y ) follows:
y\x
0
SS3657 Lecture 8
Bivariate r.v.
Joint cdf and pdf for continuous r.v.
examples
Textbook coverage: Chap 3.3
1/11
The Joint Cumulative Distribution Function (cdf)
For all possible real numbers x and y,
F (x, y) = P (X x, Y y) = P (X, Y ) (, x](, y])
In gene
SS3657 Lecture 7
Bivariate random vector (r.v.)
Joint distribution
Marginal distribution
Joint pmf and cdf of a bivariate discrete r.v.
Textbook coverage: Chap 3.2
1/10
Example 1 Toss 4 fair coins
X = number of Hs
(successes) in the rst 3
coin tosses
Y =
SS3657 Lecture 6
Transformations or functions of a r.v.
Textbook coverage: Chap 2.3
1/14
Transformation Y = g(X)
Consider a transformation Y = g(X), where g is a function
g : D R.
The domain D of g is typically the reals in R, but it must at
least contain
SS3657 Lecture 4
The denition of random variables
The properties of cdf F ()
Discrete r.v.s
cdf and pmf of a discrete r.v.
Textbook coverage: Chap 2.1
1/17
The denition of random variable (rv)
Motivation: why do we need random variable if we already
have