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ST305 Statistical Inference

创建日期：2024-06-08 17:36:50

所属分类：统计学statistics

标签： ST305

Statistical Inference

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ST305 Statistical Inference

Why study statistics?
Collecting, analyzing, and interpreting data is growing
more important every year in nearly every field.
Many important real-world decisions hinge on conflicting
claims about “what the data show,” such as:
Does raising the minimum wage increase unemployment?
Is a new cancer drug more effective than the current
standard of care?
What are the most likely ecological or agricultural effects of
climate change, and how do we mitigate them?
Interpreting data will also increasingly become part of your
personal decision making.
Knowledge on Statistics will empower us to be an active
participant in the data-driven arguments that drive
decisions and shape our understanding of the world.
2
Chapter 1 Probability Theory
“You can, for example, never foretell what any one man will
do, but you can say with precision what an average
number will be up to. Individuals vary, but percentages
remain constant. So says the statistician."
Sherlock Holmes
The signs of Four
3
Overview
Probability theory
the foundation upon which all of statistics is built
has a long and rich history, dating back to at least to the
17th century
The aims of this chapter:
to outline some of the basic ideas that are fundamental to
the study of statistics
not to give a thorough introduction to probability theory
4
Set theory
Sample space S: the set of all possible outcomes of an
experiment
Example 1: An experiment consists of flipping two coins.
The sample space
S = {(h,h), (h, t), (t ,h), (t , t)},
where t = tail and h = head .
Event: any subset of the sample space
An event is a set consisting of some possible outcomes of
the experiment.
Example 1: An experiment consists of flipping two coins.
The sample space
S = {(h,h), (h, t), (t ,h), (t , t)},
E = {(h,h), (h, t)} is an event (the event that a head
appears on the first coin).
5
Exercise: An experiment consists of flipping one coin. What is
the sample space? List one event.
6
Union
Union of events E and F : E ∪ F
E ∪ F : the event that consists of all outcomes that are
either in E
or in F
or in both E and F
Venn diagram
Example 1: Suppose E = {(h,h), (h, t)} and
F = {(t ,h), (h,h)}.
Then E ∪ F = {(h,h), (t ,h), (h,h)}.
The union, ∪∞n=1En: the event that consists of all outcomes
that are in at least one of the sets, E1, · · · , En.
7
Intersection
Intersection of events E and F : E ∩ F or EF
E ∩ F : the event that consists of all outcomes that are
both in E and in F
Venn diagram
Example 1: Suppose E = {(h,h), (h, t)} and
F = {(t ,h), (h,h)}.
Then E ∩ F = {(h,h)}.
The intersection, ∩∞n=1En: the event that consists of those
outcomes that are in all of the events, E1, · · · , En.
If E ∩ F does not contain any outcomes, it is the null event
and is denoted by ∅
8
Complementation
Complementation. Ac , the complement of A, is the set of
all elements that are not in A:
Ac = {x ; x ∈ A}
9
Exercise
Exercise: Suppose A = {(h,h), (h, t)}, B = {(t ,h)} and
C = {(h,h), (h, t), (t , t)}. Find A ∩ B, A ∪ C, A ∩ C and Cc .
10
Some rules for the operations of events
Commutative laws
E ∪ F = F ∪ E Exercise Show this is by using Venn diagrams
E ∩ F = F ∩ E Exercise Show this is by using Venn diagrams
Associative laws
E ∪ F ∪G = E ∪ (F ∪G)
E ∩ F ∩G = E ∩ (F ∩G)
Distributive laws:
(E ∪F )∩G = (E ∩G)∪(F ∩G) Exercise Show this is by using Venn diagrams
(E ∩F )∩G = (E ∩G)∩(F ∩G) Exercise Show this is by using Venn diagrams
11
DeMorgan’s laws
DeMorgan’s laws:
a) (∪ni=1Ei)c = ∩ni=1Eci
b) (∩ni=1Ei)c = ∪ni=1Eci
Proof to a):
Suppose x ∈ (∪ni=1Ei)c
=⇒ x /∈ ∪ni=1Ei
=⇒ x /∈ Ei for all i = 1,2, · · ·
=⇒ x ∈ Eci for all i = 1,2, · · ·
=⇒ x ∈ ∩ni=1Eci for all i = 1,2, · · ·
12
Disjoint (mutually exclusive) and partition
If E ∩ F = ∅, then E and F are said to be disjoint or
mutually exclusive
The events E1, E2,. . . are pairwise disjoint (or mutually
exclusive) if Ei ∩ Ej = ∅ for all i ̸= j .
If the events E1, E2,. . . are pairwise disjoint (or mutually
exclusive), then the collection of E1, E2,. . . forms a
partition of S.
Partitions are useful as it allows us to divided the smaple
space into small, non-overlapping pieces.
Example: Define Ei = [i , i + 1).
Then E1, E2,. . . are pairwise disjoint
E1, E2,. . . forms a partition of [0,∞).
13
Sigma algebra (or Borel field)
Let B be a collection of subsets of S. It is a Sigma algebra
(or Borel field) if all the following hold:
a. ∅ ∈ B
b. B is closed under complementation: If A ∈ B, then Ac ∈ B
b. B is closed under countable unions: If A1, A2, . . . ∈ B, then
∪∞i Ai ∈ B.
Example 1: {0,S}
Example 2: Let S = (−∞,∞) and B be a set that contains
all sets of the form [a,b], (a,b], (a,b) and [a,b).
14
Probability function
Given a sample space S and an associated sigma lagebra
B, a probability function P is a function with domain B
such that
Axiom 1: P(E) ≥ 0 for all E ∈ B
Axiom 2: P(S) = 1
Axiom 3: For any sequence of pairwise disjoint events, E1,
E2, . . . (EiEj = ∅ when i ̸= j),
P
(∞⋃
i=1
Ei
)
=
∞∑
i=1
P(Ei)
The three properties given above are also referred to as
the Kolmogorov Axioms (after A. Kolmogorov, one of the
fathers of probability theory).
Any function P that satisfies the Axioms of Probability is
called a probability function.
15
Example
Suppose a die (with 6 sides) is rolled and that all 6 sides are
equally likely to appear. What is the probability of rolling an
even number?
Answer.
The event of rolling even number: {2,4,6}
We have
P({1}) = P({2}) = P({3}) = P({4}) = P({5}) = P({6}) = 1
6
.
Note {2}, {4} and {6} is a sequence of mutually exclusive
events.
So by Axiom 3, the prob. of rolling an even number:
P({2,4,6}) = P({2}∪{4}∪{6}) = P({2})+P({4})+P({6}) = 1
2
.
16
A common method of defining a legitimate
probability function
Theorem: Let S = {s1, . . . , sn} be a finite set. Let B be any
sigma algebra of subsets of S. Let p1, . . . ,pn be
nonnegative numbers with
∑n
i=1 pi = 1. For any A ∈ B,
define P(A) by
P(A) =
∑
i:si∈A
pi .
(The sum over an empty set is defined to be 0 .)
Then P is a probability function on B.
This remains true if S = {s1, s2, . . .} is a countable set.
(See next slide for a proof)
17
Proof:
For any A ∈ B,P(A) =∑{i:si∈A} pi ≥ 0
=⇒ Axiom 1 is true.
P(S) =
∑
{i:si∈S}
pi =
n∑
i=1
pi = 1
=⇒ Axiom 2 is true.
Let A1, . . . ,Ak denote pairwise disjoint events. ( B contains only
a finite number of sets, so we need consider only finite disjoint
unions.) Then,
P
(
k⋃
i=1
Ai
)
=
∑
{j:sj∈∪ki=1Ai}
pj =
k∑
i=1
∑
{j:sj∈Ai}
pj =
k∑
i=1
P (Ai)
=⇒ Axiom 3 is true.
18
Some properties
P(∅) = 0
P(E) ≤ 1
P(Ec) = 1− P(E), or equivalently, P(E) + P(Ec) = 1.
If E ⊂ F , then P(E) ≤ P(F ).
Proof.
Note F = (E ∪ Ec)F = (EF ) ∪ (EcF ) = E ∪ (EcF )
(ExerciseUse Venn diagram to show this.)
Note Ec and F are mutually exclusive
By Axiom 3:
P(F ) = P(E) + P(EcF ) ≤ P(E)
(since P(EcF ) ≥ 0 by Axiom 1)
19
Some properties
P(E ∩ F c) = P(E)− P(E ∩ F ).
Proof.
E = (E ∩ F ) ∪ (E ∩ F c) (Exercise Show this in Venn
diagrams.)
E ∩ F and E ∩ F c are mutually exclusive
P(E) = P(E ∩ F ) + P(E ∩ F c)
20
Some properties
P(E ∪ F ) = P(E) + P(F )− P(EF ).
Proof.
E ∪ F = S(E ∪ F ) = (E ∪ Ec)(E ∪ F ) = E ∪ (EcF )
E and EcF mutually exclusive (by Axiom 3) =⇒
P(E ∪ F ) = P(E ∪ (EcF )) = P(E) + P(EcF )
Note F = (EF ) ∪ (EcF ) (by Axiom 3) =⇒
P(F ) = P(EF ) + P(EcF ), i.e., P(EcF ) = P(F )− P(EF )
21
Some properties
P(A) =
∑∞
i=1 P(A ∩ Ci) for any partition C1,C2, . . .
Proof.
A = A ∩ S = A ∩ (
∞⋂
i=1
Ci) =
∞⋃
i=1
(A ∩ Ci)
where the last equality follows from the Distributive Law.
Thus,
P(A) = P
(∞⋃
i=1
(A ∩ Ci)
)
Note Ci are disjoint =⇒ the sets A ∩ Ci are also disjoint
(Exercise Show this in Venn diagrams.)
P(A) = P
(∞⋃
i=1
(A ∩ Ci)
)
=
∞∑
i=1
P (A ∩ Ci)
22
Boole’s inequality
Boole’s inequality: P(E1 ∪ E2 ∪ · · · ∪ En) ≤
∑n
i=1 P(Ei).
Proof.
Construct a disjoint collection A∗1,A
∗
2, . . ., with the property that
∪∞i=1A∗i = ∪∞i=1Ai in the following way:
A∗1 = A1, A
∗
i = Ai\
i−1⋃
j=1
Aj
 , i = 2,3, . . .
Notation: A\B = A ∩ Bc.

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EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 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