Recap of Basic Probability Elements of basic probability theory - PowerPoint PPT Presentation

02407 Stochastic Processes Elements of basic probability theory Stochastic experiments Elements of basic Recap of Basic Probability Elements of basic ■ probability theory probability theory The probability triple (Ω , F , P ) : Why recap probability Why recap probability ■ theory? theory? Theory The set-up of The set-up of ◆ Ω : The sample space, ω ∈ Ω probability theory probability theory Conditional Conditional Uffe Høgsbro Thygesen ◆ F : The set of events, A ∈ F ⇒ A ⊂ Ω probabilities probabilities Stochastic variables Informatics and Mathematical Modelling Stochastic variables ◆ P : The probability measure, A ∈ F ⇒ P ( A ) ∈ [0 , 1] F X , the cumulated F X , the cumulated Technical University of Denmark distribution function distribution function Random variables (cdf) 2800 Kgs. Lyngby – Denmark (cdf) ■ Discrete and Discrete and Email: uht@imm.dtu.dk Distribution functions continuous variables continuous variables ■ Conditional Conditional expectation expectation Conditioning ■ The Bernoulli process The Bernoulli process Exercises/problems Exercises/problems 1 / 39 2 / 39 Why recap probability theory? The set-up of probability theory Elements of basic ■ Stochastic processes is applied probability Elements of basic We perform a stochastic experiment. probability theory probability theory A firm understanding of probability (as taught in e.g. 02405) will We use ω to denote the outcome. Why recap probability ■ Why recap probability theory? theory? The sample space Ω is the set of all possible outcomes. get you far The set-up of The set-up of probability theory probability theory We need a more solid basis than most students develop in e.g. ■ Conditional Conditional Ω probabilities probabilities 02405. Stochastic variables Stochastic variables F X , the cumulated F X , the cumulated distribution function distribution function What to recap? (cdf) (cdf) ω Discrete and Discrete and The concepts are most important: What is a stochastic variable, continuous variables continuous variables Conditional Conditional what is conditioning, etc. expectation expectation The Bernoulli process Specific models and formulas: That a binomial distribution appears The Bernoulli process Exercises/problems Exercises/problems as the sum of Bernoulli variates, etc. 3 / 39 4 / 39

The sample space Ω Events Ω can be a very simple set, e.g. Elements of basic Elements of basic Events are sets of outcomes/subsets of Ω probability theory probability theory Events correspond to statements about the outcome. Why recap probability Why recap probability { H, T } (tossing a coin a.k.a. Bernouilli experiment) ■ theory? theory? The set-up of The set-up of { 1 , 2 , 3 , 4 , 5 , 6 } (throwing a die once). ■ Ω probability theory probability theory For a die thrown once, the A Conditional Conditional N (typical for single discrete stochastic variables) ■ event probabilities probabilities R d (typical for multivariate continuous stochastic variables) Stochastic variables Stochastic variables ■ F X , the cumulated F X , the cumulated distribution function distribution function ω A = { 1 , 2 , 3 } or a more complicated set, e.g. (cdf) (cdf) Discrete and Discrete and The set of all functions R �→ R d with some regularity properties. continuous variables continuous variables corresponds to the statement ■ Conditional Conditional expectation expectation “the die showed no more than Often we will not need to specify what Ω is. The Bernoulli process The Bernoulli process three”. Exercises/problems Exercises/problems 5 / 39 6 / 39 Probability Logical operators as set operators Elements of basic A Probability is a set measure of an event Elements of basic An important question: Which events are “measurable”, i.e. have a probability theory probability theory If A is an event, then probability assigned to them? Why recap probability Why recap probability theory? theory? We want our usual logical reasoning to work! The set-up of The set-up of probability theory probability theory P ( A ) So: If A and B are legal statements, represented by measurable Conditional Conditional probabilities probabilities subsets of Ω , then so are Stochastic variables is the probability that the event A occurs in the stochastic Stochastic variables F X , the cumulated F X , the cumulated Not A , i.e. A c = Ω \ A distribution function distribution function ■ experiment - a number between 0 and 1. (cdf) (cdf) A or B , i.e. A ∪ B . (What exactly does this mean ? C.f. G&S p 5, and appendix III) ■ Discrete and Discrete and continuous variables continuous variables Regardless of interpretation, we can pose simle conditions for Conditional Conditional expectation expectation mathematical consistency . The Bernoulli process The Bernoulli process Exercises/problems Exercises/problems 7 / 39 8 / 39

Parallels between statements and sets An infinite, but countable, number of statements For the Bernoulli experiment, we need statements like Elements of basic Elements of basic Set Statement probability theory probability theory Why recap probability Why recap probability A “The event A occured” ( ω ∈ A ) theory? theory? At least one experiment shows heads A c The set-up of Not A The set-up of probability theory probability theory A ∩ B A and B Conditional Conditional or probabilities probabilities A ∪ B A or B Stochastic variables Stochastic variables F X , the cumulated F X , the cumulated In the long run, every other experiment shows heads. ( A ∪ B ) \ ( A ∩ B ) A exclusive-or B distribution function distribution function (cdf) (cdf) Discrete and Discrete and So: If A i are events for i ∈ N , then so is ∪ i ∈ N A i . continuous variables continuous variables See also table 1.1 in Grimmett & Stirzaker, page 3 Conditional Conditional expectation expectation The Bernoulli process The Bernoulli process Exercises/problems Exercises/problems 9 / 39 10 / 39 All events considered form a σ -field F (Trivial) examples of σ -fields Elements of basic Definition: Elements of basic 1. F = {∅ , Ω } probability theory probability theory This is the deterministic case: All statements are either true Why recap probability Why recap probability The empty set is an event, ∅ ∈ F theory? 1. theory? ( ∀ ω ) or false ( ∀ ω ). The set-up of The set-up of probability theory 2. Given a countable set of events A 1 , A 2 , . . . , its union is also an probability theory F = {∅ , A, A c , Ω } 2. Conditional Conditional event, ∪ i ∈ N A i ∈ F probabilities probabilities This corresponds to the Bernoulli experiment or tossing a coin: Stochastic variables Stochastic variables If A is an event, then so is the complementary set A c . 3. F X , the cumulated F X , the cumulated The event A corresponds to “heads”. distribution function distribution function F = 2 Ω = set of all subsets of Ω . (cdf) (cdf) 3. Discrete and Discrete and continuous variables continuous variables Conditional Conditional When Ω is finite or enumerable, we can actually work with 2 Ω ; expectation expectation The Bernoulli process The Bernoulli process otherwise not. Exercises/problems Exercises/problems 11 / 39 12 / 39