Statistics 131
MVSupranes
PARAMETRIC POINT ESTIMATION
2.1 THE PROBLEM OF POINT ESTIMATION: INTRODUCTION & DEFINITIONS
Given a parametric statistical problem specified by (, , ), where the class consists of all
the possible distributions of the random samp
Statistics 131
MVSupranes
PARAMETRIC HYPOTHESIS TESTING
4.4.THE PROBLEM OF HYPOTHESIS TESTING: INTRODUCTION &
DEFINITIONS
This chapter presents the most basic ideas concerning hypothesis testing in the classical setting. We
introduce certain tools that ca
Statistics 131
MVSupranes
Handout #0: REVIEW OF PROBABILITY THEORY
0.1 JOINT AND MARGINAL DISTRIBUTIONS
1. Let , A , P be a probability space. A random vector X 1 , X 2 ,
, X k ' is a function with
domain and counterdomain , that is ( X 1 , X 2 ,., X k )'
8
Statistics 122: Probability Theory II
1.
JOINT AND MARGINAL DISTRIBUTIONS
In Probability Theory I, we formulate the concept of a (real) random variable and
describe the probabilistic behavior of this random variable by the distributions that it induces
37
Statistics 122: Probability Theory II
3.
EXPECTATION OF SEVERAL RANDOM VARIABLES
As in Probability Theory I, the interest in most situations lies not on the actual
distribution of a random vector, but rather on a number of summary measures, which conve
Statistics 131
MVSupranes
PARAMETRIC CONFIDENCE INTERVAL ESTIMATION
3.1 THE PROBLEM OF INTERVAL ESTIMATION: INTRODUCTION & DEFINITIONS
For some inferential problems, we prefer an interval estimate to a point estimate for any of
the following reasons:
1. T
Statistics 122: Probability Theory II
4.
55
DISTRIBUTIONS OF FUNCTIONS OF RANDOM VARIABLES
It is often the case that we know the distribution of a certain random variable X and are
interested in determining the distribution of some function of X, say g(X)
28
Statistics 122: Probability Theory II
2.
CONDITIONAL DISTRIBUTIONS AND STOCHASTIC INDEPENDENCE
In Probability Theory I, a conditional probability represents the likelihood of occurrence
of an event, given that another event has already occurred. In suc
67
Statistics 122: Probability Theory II
5.
SAMPLING AND SAMPLING DISTRIBUTIONS
5.1. Introduction
Defn: The totality of elements which are under discussion, and about which information is
desired will be called the target population.
Remark: The object of
1
Statistics 122: Probability Theory II
0. Univariate Preliminaries
0.1
Probability
Random Experiment, Sample spaces and Events
Defn: An experiment that can be repeated under similar conditions, but whose outcome
cannot be predicted in advance, even when
Problem Set 1
Stat 131
AY1617
MVS
Instructions: You may work in groups of 2-3 members. Show all steps and reasons in your proofs or
solutions. Deductions will be given to vague and/or skipped steps. Properly define notations in your
solutions. Write down
Statistics 131
MVSupranes
OVERVIEW OF SAMPLING AND SAMPLING DISTRIBUTIONS
1. The totality of elements which are under discussion, and about which information is desired
will be called the target population.
2. Given a probability space and a positive inte
Stat 117 LE 3
1 Relation from A to B : subset of A B .
15 Combination.
n
k
= C (n, k) =
n!
k!(nk)!
2 Domain of R: dom R = cfw_a A : b B, (a, b) R
16 Properties of Combination:
3 Image of R: im R = cfw_b B : a A, (a, b) R
a n0 = nn = 1
n
4 Function: a
Stat 117 LE 2
1 Definition. A is a subset of B iff , A B , written as A B or B A.
2 Properties of Set Inclusion.
a Reflexive. A,
A A.
b Transitive. If A B and B C then A C .
3 Definition. A is equal to B iff A B and B A, written as A = B .
4 Set Operation
Stat 121 LE 3
DISCRETE DISTRIBUTIONS
Distribution
pX
Notation
I cfw_1,2,.,N (x)
X DU(N )
X Be p
X Bi n, p
Discrete Uniform
Bernoulli
Binomial
X Hyp (n, M , K )
Hypergeometric
x
nx
M
I cfw_0,1,.,min(n,K )
n
Geometric BEFORE
X Po ()
X Geo p
Geometric
a
X N , 2 .
Parameters.
exp
b
e
PDF.
2
Mean R, variance > 0
2 !
x
c
22
p
2
d
Mean.
MGF.
Var.
2
2 t 2
exp t +
2
Z . N(0, 1). PDF and CDF:
!
2
Std Normal Rand Var
exp
Z (z) =
z
2
p
2
, Z (z) =
Z
x
(u) du
X
f Z=
x
g If X Po( > 0), P a < p b (b) (a)
h D
Stat 117: Logic
Rules of Inference
Only premises can be substituted with conclusions, not v.v.
Rules can only be substituted to whole lines of proof.
Abbr
Premise ()
Conclusion ()
1 Modus Ponens
MP
p q, p
q
2 Modus Tollens
MT
p q, q
p
3 Hypothetical Syllo
.
.
Stat 121 LE 2
1
Random Variables.
Economical Denition.
Given , A , P , a
rand. var. X () is a function with dom X =
, codom X = R such that r R, A r =
cfw_ |X () r A
b Borel Denition. Given , A , P , a rand.
var. X () is a function with dom X = ,
cod
STATISTICS 143: INTRODUCTION TO SURVEY OPERATIONS
Open Spaces
A Survey on the Awareness and the Need for Recreational
Facilities, Park, Playgrounds and other open spaces for a
better and safer community
EVE CHRISTINE D. YAP
I.
Background of the Study
Nowa
Eve Christine Yap
201231042
Stat 149
A Reaction Paper to
A Sequential Cognitive Diagnosis Model for Polytomous Response
By Dr. Jimmy de la Torre
Many of the concepts are new to me so I was not able to catch up with the discussion
itself. However, along th
Eve Christine Yap
201231042
Stat 149
A Reaction Paper to
A Sequential Cognitive Diagnosis Model for Polytomous Response
By Dr. Jimmy de la Torre
Many of the concepts are new to me so I was not able to catch up with the discussion
itself. However, along th
Eve Yap
Ian McKeague : Steins Method , MIW and QM
Stat 149
Before anything else, the talk was very technical so there were limited portions of the talk
that I was able to comprehend. According to Sir McKeague, Quantum Mechanics is inherently
probabilistic
S T A T I S T I C S 1 3 1 N O T E S
OPTIMAL PROPERTIES OF POINT ESTIMATORS
CONSISTENCY
o MSE-consistent
1. Check if the estimator is unbiased.
2. If yes, get its variance.
3. If not, get its MSE.
4. Take the limit as n approaches infinity of the varianc
UP School of Statistics Student Council
Education and Research
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S131_Reviewer_002
Parametric Statistical Inference
Statistics 131
2nd Exam Reviewer-Summary
Some Notations
X
= (X 1
UP School of Statistics Student Council
Education and Research
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S131_Reviewer_001
Parametric Statistical Inference
Statistics 131
1st Exam Reviewer-Summary
Distribution of the ran
_
S131_REV_001
Statistics 131
A Review of Statistics 121 and 122
I.
Parametric Statistical Inference
Write True if the statement is always correct. Otherwise, write False.
1. If the range of the random variable X is the set of all possible multiples of 12
Stat 101 1st Exam Review Problems
1. Classify whether the following belongs to the area of descriptive statistics or inferential statistics:
a. As a result of recent cutbacks by the oil-producing nations, we can expect the price of gasoline to double in
t