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STAT 430 (1)
Lecture

Stat 450 - Review of Probability and Distribution

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Department
Statistics
Course
STAT 430
Professor
Jinko Graham
Semester
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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