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Lecture7

Lecture7 - Variance Denition Let X be a numerically valued...

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Variance 10/13/2005

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Definition Let X be a numerically valued random variable with expected value μ = E ( X ) . Then the variance of X , denoted by V ( X ) , is V ( X ) = E (( X - μ ) 2 ) . 1
Standard Deviation The standard deviation of X , denoted by D ( X ) , is D ( X ) = p V ( X ) . We often write σ for D ( X ) and σ 2 for V ( X ) . 2

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Example Consider one roll of a die. Let X be the number that turns up. μ = E ( X ) = 1 1 6 · + 2 1 6 · + 3 1 6 · + 4 1 6 · + 5 1 6 · + 6 1 6 · = 7 2 . 3
x m ( x ) ( x - 7 / 2) 2 1 1/6 25/4 2 1/6 9/4 3 1/6 1/4 4 1/6 1/4 5 1/6 9/4 6 1/6 25/4 V ( X ) = 1 6 25 4 + 9 4 + 1 4 + 1 4 + 9 4 + 25 4 = 35 12 , 4

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Calculation of Variance Theorem. If X is any random variable with E ( X ) = μ , then V ( X ) = E ( X 2 ) - μ 2 . 5
Example (cont) E ( X 2 ) = 1 1 6 · + 4 1 6 · + 9 1 6 · + 16 1 6 · + 25 1 6 · + 36 1 6 · = 91 6 , and, V ( X ) = E ( X 2 ) - μ 2 = 91 6 - 7 2 · 2 = 35 12 . 6

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Properties of Variance Theorem. If X is any random variable and c is any constant, then V ( cX ) = c 2 V ( X ) and V ( X + c ) = V ( X ) .
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Lecture7 - Variance Denition Let X be a numerically valued...

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