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Unformatted text preview: AMS 410 Final Review AMS 410 Final Review Ning SUN December 8, 2009 ◮ *: easy to make mistakes ◮ **: not requested in the final, but included in exam P’s syllabus AMS 410 Final Review Table of Contents General Probability Union, Intersection and Complement Bayes’ Formula Univariate Distribution General Distribution: One Random Variable Specific Distributions: Three Discrete Specific Distributions: Three Continuous Multivariate Distribution General Distribution: Two Random Variable Some Important Topics Variance and Covariance Moment Generating Function Independence Normal and CLT Transformation Final Exam Information AMS 410 Final Review General Probability General Probability 1. Union, Intersection and Complement 2. Bayes’ Formula AMS 410 Final Review General Probability Union, Intersection and Complement Union, Intersection and Complement ◮ The most important formulas: ◮ P [ A ∪ B ∪ C ] = P [ A ]+ P [ B ]+ P [ C ] P [ A ∩ B ] P [ A ∩ C ] P [ B ∩ C ]+ P [ A ∩ B ∩ C ] ◮ P [ A ∪ B ] = P [ A ] + P [ B ] P [ A ∩ B ] ◮ P [ A ′ ] = 1 P [ A ], P [( A ∩ B ) ′ ] = P [ A ′ ∪ B ′ ], P [( A ∪ B ) ′ ] = P [ A ′ ∩ B ′ ] ◮ P [ A ∩ B ] = P [ A  B ] P [ B ] ◮ Sometimes the probabilities are not given directly. ◮ Sometimes independence is given. You can use P [ A ∩ B ] = P [ A ] P [ B ]. ◮ Sometimes you need to calculate # of elements in a set ◮ Hints: Remember the formula, get familiar with Venn Diagram (set and probability). AMS 410 Final Review General Probability Union, Intersection and Complement Examples ◮ A doctor is studying the relationship between blood pressure and heartbeat abnormalities in her patients. She tests a random sample of her patients and notes their blood pressures (high, low, or normal)and their heartbeats (regular or irregular). She finds that: 1. 14% have high blood pressure. 2. 22% have low blood pressure. 3. 15% have an irregular heartbeat. 4. Of those with an irregular heartbeat, onethird have high blood pressure. 5. OF those with normal blood pressure, oneeighth have an irregular heartbeat. What portion of the patients selected have a regular heartbeat and low blood pressure? AMS 410 Final Review General Probability Union, Intersection and Complement Examples ◮ An auto insurance company has 10,000 policyholders. Each policyholder is classified as: ◮ young or old; ◮ male or female; ◮ married or single. Of these policyholders, 3,000 are young, 4,600 are male, and 7,000 are married. The policyholders can also be classified as 1,320 young males, 3,010 married males, and 1,400 young married persons. Finally, 600 of the policyholders are young married males. How many of the company’s policyholders are young, female, and single? AMS 410 Final Review General Probability Bayes’ Formula Bayes’ Formula ◮ The most important formulas: ◮ Law of Total Probability P [ B ] = P [ B ∩ A ]+ P [ B ∩ A ′ ] = P [ B  A ] · P [ A ]+ P [ B  A ′ ] ·...
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 Fall '08
 Yang,Y
 Probability theory, insurance policy

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