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STAT 430
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Jinko Graham
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Lecture

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Statistics

STAT 430

Jinko Graham

Fall

Description

Statistics 450: Review of Probability and Distributions
The main purpose of reviewing this pre-requisite material is to establish notation to be used throughout
the rest of the course; it will also serve as an overview of material you are responsible for going in to the
course. Topics are: (1) Set theory review, (2) Elementary probability theory, (3) Random variables, (4)
Cumulative distribution functions, (5) Discrete random variables, (6) Continuous random variables, (7)
Expections, (8) Transformations. This handout contains sections for topics 1-7 which people are expected
to review on their own. Well brie
y review topic 8 in class.
1 Set theory review
Denition of a set: A collection of objects
Represent by a Venn diagram
c
A [ B A \ B A and its \complement" A
$
A
$ $
& %
$ $
A A
& % & %
B B
& % & % A
A B AnB or \A take away B"
$
$
B $
$
A A B
&& %
& %
& %
Some set operations
1. Union: A [ B = fx : x 2 A or x 2 Bg
Read \ set of all x such that x is in A or x is in B"
2. Intersection: A \ B = fx : x 2 A and x 2 Bg. Two sets are said to be disjoint if A \ B = ;
(the empty set).
3. subset: A B if and only if (i) x 2 A ) x 2 B 8x () is \implies", 8 is \for all")
4. complement: A = fx : x 62 Ag
5. dierence: AnB = fx : x 2 A and x 62 Bg = A \ B
Set identities
1 c c c c c c
I1 DeMorgans laws: (A [ B) = A \ B and (A \ B) = A [ B
I2 Associative law for [: (A [ B) [ C = A [ (B [ C)
I3 Commutative law for [: A [ B = B [ A and A [ B [ C = A [ C [ B = B [ A [ C, etc.
I4 Distributive law for \:
1. A \ (B [ C) = (A \ B) [ (A \ C)
2. A [ (B \ C) = (A [ B) \ (A [ C)
2 Elementary probability theory
Random experiment: cannot predict outcomes, but all possible outcomes are clear and there is a
predictable behaviour in a large number of repetitions
Sample space S: All possible outcomes of a random experiment
Events and the event space
{ A subset of the sample space S is called an event
{ Denition: The set of all events associated with a given experiment is called the event space,
denoted by B: B = fA : A Sg
Probability functions
{ Probability functions P() have the event space B as their domain and [0;1] as their range; i.e.
P : B ! [0;1]
{ Denition: A probability function is a set function from B to [0;1] satisfying the following rules
or axioms:
A1 P(A) 0 8 A 2 B
A2 P(S) = 1
A3 If A 1A 2A 3::: is a sequence of mutually exclusive events in B (i.e. j \A = ; for i 6= j),
then
1 ! 1
[ X
P A i = P(A i
i=1 i=1
{ Denition: The triplet (S;B;P) is called a probability space
{ Some properties of probability
P1 P(;) = 0
P2 If Ai\ A j ; for i 6= j, then P(1 [ [nA ) = P(1 ) + + PnA ); special case of (A3)
P3 P(A ) = 1 P(A).
P4 P(AnB) = P(A \ B ) = P(A) P(A \ B).
P5 If A B, then P(A) P(B).
P6 P(A [ B) = P(A) + P(B) P(A \ B), Called the \inclusion-exclusion formula" and the
principle holds more generally.
{ Inclusion-exclusion formula:
X X X
P(A 1 A 2 [ A n = P(A i P(A i A )j+ P(A i A \jA ) k
i 1i

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