9_21_11

9_21_11 - Variance Discrete Uniform Distribution...

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Unformatted text preview: Variance, Discrete Uniform Distribution, Permutations and Combinations Instructor: Andrew Liu September 21, 2011 Textbook sections: 3-4, 3-5, 2-1.4 Review Variance: 2-dice-throw example 0 1/36 1/18 1/12 1/9 5/36 1/6 7/36 2 3 4 5 6 7 8 9 10 11 12 Probability Range of Random Variable Probability Mass Function E ( X ) = 2 * 1 36 +3 * 1 18 +4 * 1 12 + ··· +12 * 1 36 = 7 ( make sense ?) . Textbook sections: 3-4, 3-5, 2-1.4 Review Variance: 2-dice-throw example 0 1/36 1/18 1/12 1/9 5/36 1/6 7/36 2 3 4 5 6 7 8 9 10 11 12 Probability Range of Random Variable Probability Mass Function E ( X ) = 2 * 1 36 +3 * 1 18 +4 * 1 12 + ··· +12 * 1 36 = 7 ( make sense ?) . V ( X ) = E ( X 2 )- [ E ( X )] 2 = 2 2 * 1 36 + 3 2 * 1 18 + ··· + 12 2 * 1 36- 7 2 ≈ 5 . 83 . Textbook sections: 3-4, 3-5, 2-1.4 Review Variance: 2-dice-throw example 0 1/36 1/18 1/12 1/9 5/36 1/6 7/36 2 3 4 5 6 7 8 9 10 11 12 Probability Range of Random Variable Probability Mass Function E ( X ) = 2 * 1 36 +3 * 1 18 +4 * 1 12 + ··· +12 * 1 36 = 7 ( make sense ?) . V ( X ) = E ( X 2 )- [ E ( X )] 2 = 2 2 * 1 36 + 3 2 * 1 18 + ··· + 12 2 * 1 36- 7 2 ≈ 5 . 83 . σ ( X ) ≈ 2 . 42 . Textbook sections: 3-4, 3-5, 2-1.4 Examples (cont.) Given a random variable X . Suppose its probability mass function is given as follows. f ( x ) = ( 8 7 )( 1 2 ) x , x = 1 , 2 , 3 . Calculate V ( X ). Textbook sections: 3-4, 3-5, 2-1.4 Examples (cont.) Given a random variable X . Suppose its probability mass function is given as follows. f ( x ) = ( 8 7 )( 1 2 ) x , x = 1 , 2 , 3 . Calculate V ( X ). Solution: (Just use the formula of variance!) E ( X ) = X x xf ( x ) = 1 * ( 8 7 )( 1 2 ) + 2 * ( 8 7 )( 1 2 ) 2 + 3 * ( 8 7 )( 1 2 ) 3 ≈ 1 . 57 . V ( X ) = E ( X 2 )- [ E ( X )] 2 = Textbook sections: 3-4, 3-5, 2-1.4 Examples (cont.) Given a random variable X . Suppose its probability mass function is given as follows....
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9_21_11 - Variance Discrete Uniform Distribution...

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