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Example_class_3_handout-solution

# Example_class_3_handout-solution - THE UNIVERSITY OF HONG...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 3 Review Probability (a) Law of Total Probability (i) If 0 < P ( B ) < 1, then P ( A ) = P ( A | B ) P ( B ) + P ( A | B c ) P ( B c ) for any A. (ii) If B 1 , B 2 , ..., B k are mutually exclusive and exhaustive events (i.e. a partition of the sample space), then for any event A, P ( A ) = k X j =1 P ( A | B j ) P ( B j ) where k can also be . (b) Bayes’ Theorem (Bayes’ rule, Bayes’ law) For any two event A and B with P ( A ) > 0 and P ( B ) > 0, P ( B | A ) = P ( A | B ) P ( B ) P ( A ) (c) Bayes’ Theorem If B 1 , B 2 , ..., B k are mutually exclusive and exhaustive events (i.e. a partition of the sample space), and A is any event with P ( A ) > 0 , then for any B j , P ( B j | A ) = P ( A | B j ) P ( B j ) P ( A ) = P ( B j ) P ( A | B j ) k i =1 P ( B i ) P ( A | B i ) where k can also be . Random variables (a) Basics - Random variable is a measurable function between a Sample Space (Domain) and State Space (Range). - A function (with some requirement) X : Ω 3 ω 7→ X ( ω ) X (Ω) defined on the sample space Ω = { ω } is called a random variable. 1

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(b) Distribution- Law of a Random Variable’s Dance There is a law governing how any random variable to be observed in its state space. The law is a probabilistic one, called the probability distribution of a random variable. There are two qualifications for any real-valued function f ( x ) to be a probability density/mass function. (a) f ( x ) > 0 for any x X (Ω) (b) ´ x X (Ω) f ( x ) dx = 1 or x X (Ω) f ( x ) dx = 1 (c) Real valued random variables There are three groups of real-valued random variables: (a) Discrete random variables (its state space is a discrete subset in R ) (b) Continuous random variables (its state space is a continuous subset in R ) (c) Partially Discrete and Partially Continuous Random Variables The Distribution of a Discrete/Continuous Random Variable is called its Probability Mass/Density Function because of a superficial difference in mathematical treatments and graphical repre- sentation. Problems Problem 1. A small plane have gone down, and the search is organized into three regions. Starting with the likeliest, they are: Region Initial Chance Plane is there Chance of Being Overlooked in the Search Mountains 0.5 0.3 Praire 0.3 0.2 Sea 0.2 0.9 The last column gives the chance that if the plane is there, it will not be found. For example, if it went down at sea, there is 90% chance it will have disappeared, or otherwise not be found. Since 2
the pilot is not equipped to long survive a crash in the mountains, it is particularly important to determine the chance that the plane went down in the mountains.

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Example_class_3_handout-solution - THE UNIVERSITY OF HONG...

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