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Unformatted text preview: Variance 10/13/2005 Definition Let X be a numerically valued random variable with expected value = E ( X ) . Then the variance of X , denoted by V ( X ) , is V ( X ) = E (( X ) 2 ) . 1 Standard Deviation The standard deviation of X , denoted by D ( X ) , is D ( X ) = p V ( X ) . We often write for D ( X ) and 2 for V ( X ) . 2 Example Consider one roll of a die. Let X be the number that turns up. = 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 Calculation of Variance Theorem. If X is any random variable with E ( X ) = , then V ( X ) = E ( X 2 ) 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 ) 2 = 91 6 7 2 2 = 35 12 ....
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This note was uploaded on 07/16/2010 for the course MATH 20 taught by Professor Ionescu during the Fall '05 term at Dartmouth.
 Fall '05
 Ionescu
 Probability, Standard Deviation, Variance

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