LECTURE 9Continuous r.v.’s and PDFsSummary of ConceptsJoint PDF (1)Joint PDF (2)ConditioningExample: Stick-Breaking (1)Stick-Breaking (2)Stick-Breaking (3)LECTURE 9• Readings: Section 3.4-3.5Lecture outline•PDF: Review•Multiple random variables– Conditioning– Independence•ExamplesContinuous r.v.’s and PDFsSummary of ConceptsJoint PDF (1)•Interpretation:• Expectation:Joint PDF (2)• From the joint to the marginal:•X and Y are called independent iff:Conditioning• Recall, again:• By analogy:• Thus, the definition:• Conditioning is a “section” ofthe joint PDF, normalized.• Independence gives:Example: Stick-Breaking (1)• Break a stick of length twice:– X: first break point, chosen uniformly between 0 and .– Y: second break point, chosen (given X=x) uniformly from 0 to x.Stick-Breaking (2)•Joint PDF:Stick-Breaking (3)• Conditional Expectation of Y, given X=x:• Expectation of
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