Probability and Random Processes

Transcription

1 Probability and Random Processes Lecture 7 Conditional probability and expectation Decomposition of measures Mikael Skoglund, Probability and random processes 1/13 Conditional Probability A probability space (Ω, A, P ) An event F A with P (F ) > 0; the σ-algebra generated by F, = σ({f }) = {, F, F c, Ω} Elementary conditional probability of E A given F P (E F ) = P (E F ) P (F ) The conditional probability of E A conditioned on = the probability of E knowing which events in occurred = probability of E knowing whether F or F c occurred P (E ) = P (E F )χ F (ω) + P (E F c )χ F c(ω) a function Ω : R Mikael Skoglund, Probability and random processes 2/13

2 Note that P (E ) is a random variable on (Ω, A, P ); is -measurable; and that P ( E) = P (E )dp, A basis for generalizing P (E ) to conditioning on arbitrary σ-algebras Mikael Skoglund, Probability and random processes 3/13 iven (Ω, A, P ), E A and A, there exists a nonnegative -measurable function P (E ) such that P ( E) = P (E )dp, Also, P (E ) is unique P -a.e. Proof: Define µ E () = P ( E) for any, then µ E P and P (E ) = dµ E dp The function P (E ) is called the conditional probability of E given the probability of E knowing which events in occurred Mikael Skoglund, Probability and random processes 4/13

3 Again, for fixed and E, the entity P (E ) is a function f(ω) = P (E )(ω) on Ω Alternatively, by instead fixing and ω we get a set function m(e) = P (E )(ω), E A If m(e) is a probability measure on (Ω, A) then P (E ) is said to be regular P (E ) is in general not necessarily regular... If the space (Ω, A) is standard (more about this later in the course), then m(e) is a probability measure Mikael Skoglund, Probability and random processes 5/13 Conditioning on a Random Variable iven (Ω, A, P ) and a random variable X, let σ(x) = smallest F A such that X is (still) measurable w.r.t. F = the σ-algebra generated by X, σ(x) is exactly the class of events for which you can get to know whether they occured or not by observing X The conditional probability of E A given X is defined as P (E X) = P (E σ(x)) Mikael Skoglund, Probability and random processes 6/13

4 Signed Measure iven a measurable space (Ω, A), a signed measure ν on A is an extended real-valued function such that ν( ) = 0 for a sequence {A i } of pairwise disjoint sets in A ( ) ν A i = ν(a i ) i i (i.e., simply a measure that doesn t need to be positive) Mikael Skoglund, Probability and random processes 7/13 Radon Nikodym for Signed Measures If µ is a σ-finite measure and ν a finite signed measure on (Ω, A), and also ν µ, then there is an integrable real-valued A-measurable function f on Ω such that ν(a) = fdµ for any A A. Furthermore, f is unique µ-a.e. The function f is the Radon Nikodym derivative of ν w.r.t. µ, notation f = dν dµ A Mikael Skoglund, Probability and random processes 8/13

5 Conditional Expectation iven (Ω, A, P ), a random variable Y (with E[ Y ] < ) and A, there exists a -measurable function E[Y ] such that Y dp = E[Y ]dp, Also, the function E[Y ] is unique P -a.e. Proof: Define µ Y () = Y dp for any, then µ Y P and E[Y ] = dµ Y dp The function E[Y ] is called the conditional expectation of Y given the expectation of Y knowing which events in occurred Mikael Skoglund, Probability and random processes 9/13 Conditional Expectation vs. Probability The entity E[Y ] is a function g(ω) = E[Y ](ω) If (Ω, A) is standard, then P (E ) is regular m(e) = P (E )(ω) is a probability measure on (Ω, A) for fixed ω and. Furthermore, in this case E[Y ] = Y (u)dm(u) = Y (u)dp (u ) This interpretation for conditional expectation does not hold in general (for non-standard (Ω, A)) Mikael Skoglund, Probability and random processes 10/13

6 Mutually Singular Measures iven (Ω, A), two measures µ 1 and µ 2 are mutually singular, notation µ 1 µ 2, if there is a set E A such that µ 1 (E c ) = 0 and µ 2 (E) = 0. Lebesgue decomposition: iven a σ-finite measure space (Ω, A, µ) and an additional σ-finite measure ν on A, there exist measures ν 1 and ν 2 on A such that ν 1 µ, ν 2 µ and ν = ν 1 + ν 2. This representation is unique. Mikael Skoglund, Probability and random processes 11/13 Continuous and Discrete Measures For a measure space (Ω, A, µ) such that {x} A for all x Ω: x Ω is an atom of µ if µ({x}) > 0 µ is continuous if it has no atoms µ is discrete if there is a countable K Ω such that µ(k c ) = 0 Let (Ω, A, µ) be a σ-finite measure space and ν an additional σ-finite measure on A. Assume that {x} A for all x Ω. Then there exist measures ν ac, ν sc and ν d such that ν ac µ, ν sc µ an ν d µ ν sc is continuous and ν d is discrete ν = ν ac + ν sc + ν d, uniquely Mikael Skoglund, Probability and random processes 12/13

7 Decomposition on the Real Line Let ν be a finite Borel measure on R, then ν can be decomposed uniquely as ν = ν ac + ν sc + ν d where ν ac is absolutely continuous w.r.t. Lebesgue measure ν sc is continuous and singular w.r.t Lebesgue measure ν d is discrete Furthermore, if F ν is the distribution function of ν, then ν({x}) = F ν (x) lim F ν(x ) x x That is, if there are atoms, they are the points of discontinuity of F ν Mikael Skoglund, Probability and random processes 13/13