MA115- Statistics I
Mamikon S. Ginovyan
1
Chapter 6
Discrete Probability Distributions.
The Binomial Distribution
Mamikon Ginovyan
2
Discrete Random Variables
Basic Topics:
1. Definition of Random Variables. Two Types of
Random Variables.
2. Probability D
MA115- Statistics I
Mamikon S. Ginovyan
1
Chapter 7
Continuous Probability Distributions.
The Normal Distribution
Mamikon Ginovyan
2
Continuous Random Variables
Basic Topics:
1. Probability Distributions of Continuous
Random Variables.
2. Properties of th
MA115- MID-2 Review
Probability
Mamikon S. Ginovyan
1
Probability
Basic Titles:
1. Elements of Probability. Counting Rules. Simple
Random Sampling (Chapter 5)
2. Discrete Probability Distributions. The Binomial
Distribution (Chapter 6)
3. Continuous Proba
Time Series Analysis: Model Identication
Estimation of ACF and PACF
Ashis Gangopadhyay
Spring 2016
Ashis Gangopadhyay (BostonUniversity)
Model Identication
Spring 2016
1 / 33
Estimation: Preliminaries
In this section we will review some basic facts of par
Time Series Analysis
Forecasting
Ashis Gangopadhyay
Spring 2016
Ashis Gangopadhyay (Boston University)
Forecasting
Spring 2016
1 / 37
Forecasting
One of the main objectives of time series analysis is to utilize the past
information to forecast for future
Time Series Analysis
Parameter Estimation in ARIMA Models
Ashis Gangopadhyay
Spring 2016
Ashis Gangopadhyay (Boston University)
Parameter Estimation
Spring 2016
1 / 30
Parameter Estimation
One important aspect of time series modeling is the estimation of
Time Series Analysis
ARCH and GARCH Processes
Ashis Gangopadhyay
Spring 2016
Ashis Gangopadhyay (Boston University)
ARCH and GARCH
Spring 2016
1 / 28
Financial Time Series
Ashis Gangopadhyay (Boston University)
ARCH and GARCH
Spring 2016
2 / 28
Financial
Time Series Analysis: Characterizing Dependence
Ashis Gangopadhyay
Boston University
Spring 2016
Ashis Gangopadhyay (Boston University)
Characterizing Dependence
Spring 2016
1 / 25
Time Series Process
Denition
Let (,F ,P ) be a probability space, and let
MA585
TimeSeriesAnalysisandForecasting
Regression:AReview
Terminology: Independent and
Dependent Variables
We use independent variable x to predict
p
variable yy.
dependent
We use study hours to predict test score.
We use population of city to predict cr
Time Seies Analysis Analysis
Nonstationarity
Ashis Gangopadhyay
Spring 2016
Ashis Gangopadhyay (Boston University)
Nonstationarity
Spring 2016
1 / 47
Nonstationarity
Time Series models we have developed assumes that the data is
stationary.
Ashis Gangopadh
Time Series Analysis: Model Building
Autoregressive Moving Average Processes
Ashis Gangopadhyay
Boston University
Spring 2016
Ashis Gangopadhyay (Boston University)
ARMA Models
Spring 2016
1 / 49
Linear Process
Denition
Let et be WN (0, 2 ). A process Xt
Time Series Analysis
Probability Review
Ashis Gangopadhyay
Boston University
Spring 2016
Ashis Gangopadhyay (Boston University)
Probability Review
Spring 2016
1 / 32
Probability Space
Objective of statistical procedures are to try to model "random
experim
SOLUTIONS
IE409: Time Series Analysis
Fall 2011
Homework 1
20 September 2011
(1) (B&D 1.4) Let cfw_Zt be a sequence of independent normal random variables, each with mean 0 and
variance 2 , and let a, b, and c be constants. Which, if any, of the followin
SOLUTIONS
IE409: Time Series Analysis
Fall 2011
Homework 2
4 October 2011
(1) (B&D 1.5) Let cfw_Xt be the moving-average process of order 2 given by
Xt = Zt + Zt2 ,
where cfw_Zt is WN(0, 1).
(a) Find the autocovariance and autocorrelation functions for
Bounds:
Upper Bounds:
A set of real numbers, A is bounded above if:
a number x such that x a for every
a
in A
Such a number x is called an upper bound for A.
Entire sets of numbers (such as
and ) are not bounded above
A number x is a least upper bound (or
Theorem 10.1
If f is a constant function,
f ( x )=c , then,
f ' ( a ) =0 for all numbers a
Theorem 10.2
If f is the identity function,
f ( x )=c
then,
f ' ( a ) =1 for all numbers a
Theorem 10.3
If f and g are differentiable at
'
'
a , then
f +g
is also d
Differentiability:
Function f is differentiable at a if
lim
h0
f ( a+ h )f (a)
exists
h
f ' ( a)
This limit is denoted by
df ( x ) dy
dx
dx
Also can be represented by
Second Derivative can be represented by
f ' ( a)
f at a
is called the derivative of
f '
Theorem 7.1: Specific case of the Intermediate Value Theorem
If f is continuous on [a, b], and f ( a )< 0< f ( b ) , then
some x [ a , b ] such that f ( x )=0
Theorem 7.2: Bounded above
If f is continuous on [a, b], then
f
is bounded above on [a, b],
In
Continuous Functions:
The function f is continuous at
1.
2.
3.
f
If
a
if:
f ( a ) exist s
lim f ( x )
exists
x a
lim f ( x ) =f ( a)
x a
x
is continuous at
be continuous on
for all
x
on the interval
(a , b) , then
f is said to
(a , b)
If this statement ho
Limit:
f approaches the limit
The function
l
to
l
near
a
if we can make
f ( x)
as close
as we like by making x be close enough, but unequal to a.
In other words:
f
If
approaches a limit near a then:
> 0, some > 0
For every
For all x (as long as:
such tha
Function:
A function is a collection of pairs of numbers with the following property:
If (a,b) and (a,c) are both in the collection, then b=c
In other words:
The collection must not contain two different pairs with the same first element
Domain:
If f is a
Real Line:
The geometric picture of the set
R
Numbers are given by points
Assumed that all irrational numbers fit on the line
Uniform distance between points
If a<b then the point corresponding to a will lie left to the point corresponding
to b
|ab| is th
Summable
cfw_a n
The sequence
is summable if the sequence
cfw_sn converges,
cfw_ sn =a1 +. .+ an
Define
cfw_ sn = an
n=1
, this is called the sum of the sequence
lim
cfw_a n
n
Cauchy Criterion:
The sequence is summable if and only if:
lim a n+1 +.+a
Types of Numbers:
Natur
al
0, 1, 2, 3, 4, . or 1, 2, 3, 4, .
Integ
ers
., 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, .
Ratio
nal
a
b where a and b are integers and b is not zero
Real
Contains not only rational numbers, but also
irrational numbers(numbers with infin
Basic Properties of Numbers:
(P1)
(P2)
(P3)
(P4)
(P5)
(P8)
(P9)
(P10)
Associative law for addition
Existence of an additive identity
Existence of additive inverses
Commutative law for addition
Associative law for multiplication
Existance of a multiplicati
Convergence
cfw_f n be a sequence of functions defined on A.
Let
f
Let
f
be a function which is also defined on A
cfw_f n on A if
is called the uniform limit of
For every
> 0 there is some
n>N , then
If
We also say that
The series of
N
x in A
such that
Taylor Polynomial
n
Suppose that p ( x )=a0 +a 1 x+ .+ an x
p ( 0 )=a 0
'
p (0)=a1
p' ' ( 0 )=2 a2
pk ( 0 )=k ! ak
Now suppose that
n
p ( x )=a0 +a 1(x a)+ .+ an (xa)
k
p (a )
ak=
k!
Let
f
be a function where
ak=
f 1 ( a ) f n (a)
all exist
k
f (a)
k!
Pn
Infinite Sequences
An infinite sequence of real numbers is a function whose domain is
cfw_a n
A sequence
if for every
if
converges to
l
> 0 N such that for all natural numbers n
n> N , then
|a nl|<
In symbols:
lim an=l
n
If it doesnt converge, then it
Unit Circle:
Radius = 1
Centered on the origin
Area =
1
=2 1x 2 dx
1
Coordinates of the circle
(cos x , sin x )
Sine
sin opposite / hypotenuse
Cosine
cos adjacent / hypotenuse
Tangent
t
an
opposite / adjacent
Cotange
cot adjacent / opposite
nt
Secant
se
Critical Points:
Maximum Point:
x
A point
in A is a maximum point for
f
on A if
f
on A if
f ( x ) f ( y ) for every y A
Minimum Point:
x
A point
in A is a minimum point for
f ( x ) f ( y ) for every y A
Maximum/Minimum Value:
f ( x ) itself is called the