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9_26_11 - Binomial Distribution Geometric Distribution...

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Binomial Distribution, Geometric Distribution, Poisson Distribution Instructor: Andrew Liu September 26, 2011 Textbook sections: 3-6, 3-7, 3-9
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Review: Bernoulli Trial Similarities among the previous examples: The trials are independent Only two possible outcomes for the trial Suppose one outcome is called “success”, the other called “failure”. The probability of success (or of failure) is the same of all trials . Such a trial (not the repetition of the trials) is called a Bernoulli trial . Textbook sections: 3-6, 3-7, 3-9
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Binomial Distribution f ( x ) = n x ! p x (1 - p ) ( n - x ) , x = 0 , 1 , 2 , . . . , n . Textbook sections: 3-6, 3-7, 3-9
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Binomial Distribution f ( x ) = n x ! p x (1 - p ) ( n - x ) , x = 0 , 1 , 2 , . . . , n . Suppose that n = 10, p = 0 . 6. 0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 9 10 Probability X Figure: Probability Mass Funciton of a Binomial Random Variable Textbook sections: 3-6, 3-7, 3-9
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Binomial Distribution: Example Suppose that the probability that Robin Hood hits a bullseye with his bow is 99%. Now he enters a competition with each competitor shoots ten arrows. Probability that Robin Hood hits the bullseye 8 times? Probability that Robin Hood hits the bullseye 5 times or more? Probability that Robin Hood misses the bullseye 2 times? Textbook sections: 3-6, 3-7, 3-9
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Binomial Distribution: Mean and Variance What is E ( X ) and V ( X )? Textbook sections: 3-6, 3-7, 3-9
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Binomial Distribution: Mean and Variance What is E ( X ) and V ( X )? If X is a binomial random variable with parameters p and n , then μ = E ( X ) = np , σ 2 = V ( X ) = np (1 - p ) . Textbook sections: 3-6, 3-7, 3-9
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Binomial Distribution: Mean and Variance What is E ( X ) and V ( X )? If X is a binomial random variable with parameters p and n , then μ = E ( X ) = np , σ 2 = V ( X ) = np (1 - p ) . In the Robin Hood example, let X be the times that an arrow hits the bullseye. Textbook sections: 3-6, 3-7, 3-9
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Binomial Distribution: Mean and Variance What is E ( X ) and V ( X )? If X is a binomial random variable with parameters p and n , then μ = E ( X ) = np , σ 2 = V ( X ) = np (1 - p ) . In the Robin Hood example, let X be the times that an arrow hits the bullseye. E ( X ) = 10 * 0 . 99 = 9 . 9, Textbook sections: 3-6, 3-7, 3-9
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Binomial Distribution: Mean and Variance What is E ( X ) and V ( X )? If X is a binomial random variable with parameters p and n , then μ = E ( X ) = np , σ 2 = V ( X ) = np (1 - p ) . In the Robin Hood example, let X be the times that an arrow hits the bullseye. E ( X ) = 10 * 0 . 99 = 9 . 9, V ( X ) = 10 * 0 . 99 * (1 - 0 . 99) = 0 . 099, Textbook sections: 3-6, 3-7, 3-9
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Binomial Distribution: Mean and Variance What is E ( X ) and V ( X )? If X is a binomial random variable with parameters p and n , then μ = E ( X ) = np , σ 2 = V ( X ) = np (1 - p ) . In the Robin Hood example, let X be the times that an arrow hits the bullseye. E ( X ) = 10 * 0 . 99 = 9 . 9, V ( X ) = 10 * 0 . 99 * (1 - 0 . 99) = 0 . 099, σ = p V ( X ) 0 . 3146. Textbook sections: 3-6, 3-7, 3-9
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Binomial Distribution: Another Example Suppose that the phone lines to an airline reservation system are occupied 40% of the time. Assume that the events that the lines are occupied on successive calls are independent. For 10 calls that are placed to the airline, Textbook sections: 3-6, 3-7, 3-9
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Binomial Distribution: Another Example Suppose that the phone lines to an airline reservation system are occupied 40% of the time. Assume
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