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100AHW3 - person is tested negative what is the chance that...

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STAT 100A HWIII Due next Wed Problem 1: Prove the following two identities ( C stands for cause, E stands for effect), where { C i , i = 1 , ..., n } partition the whole sample space. (1) Rule of total probability: P ( E ) = n i =1 P ( C i ) P ( E | C i ). (2) Bayes rule: P ( C j | E ) = P ( C j ) P ( E | C j ) / n i =1 P ( C i ) P ( E | C i ). Problem 2: Suppose 1% of the population is inflicted with a particular disease. For a medical test, if a person has the disease, then 95% chance the person will be tested positive. If a person does not have the disease, then 90% chance the person will be tested negative. Using precise notation, calculate (1) The probability that a randomly selected person will be tested positive. (2) If the person is tested positive, what is the chance that he or she has the disease? If the
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Unformatted text preview: person is tested negative, what is the chance that the person has no disease? Problem 3: Suppose we roll a biased die, the probability mass function is p (1) = . 1, p (2) = . 1, p (3) = . 1, p (4) = . 2, p (5) = . 2, and p (6) = . 3. Let X be the random number we get by rolling this die. (1) Calculate Pr( X > 4). (2) Calculate E( X ). (3) Suppose the rewards for the six numbers are respectively h (1) =-$20, h (2) =-$10, h (3) = $0, h (4) = $10, h (5) = $20, and h (6) = $100. Please calculate E[ h ( X )]. (4) Please interpret E( X ) and E[ h ( X )] in terms of long run averages. 1...
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