elemprob-page11 - X ) is the standard deviation of X . The...

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Similarly we have E ( cX ) = c E X if c is a constant. These linearity results are quite hard using the first definition. It turns out there is a formula for the expectation of random variables like X 2 and e X . To see how this works, let us first look at an example. Suppose we roll a die and let X be the value that is showing. We want the expectation E X 2 . Let Y = X 2 , so that P ( Y = 1) = 1 6 , P ( Y = 4) = 1 6 , etc. and E X 2 = E Y = (1) 1 6 + (4) 1 6 + ··· + (36) 1 6 . We can also write this as E X 2 = (1 2 ) 1 6 + (2 2 ) 1 6 + ··· + (6 2 ) 1 6 , which suggests that a formula for E X 2 is x x 2 P ( X = x ). This turns out to be correct. The only possibility where things could go wrong is if more than one value of X leads to the same value of X 2 . For example, suppose P ( X = - 2) = 1 8 , P ( X = - 1) = 1 4 , P ( X = 1) = 3 8 , P ( X = 2) = 1 4 . Then if Y = X 2 , P ( Y = 1) = 5 8 and P ( Y = 4) = 3 8 . Then E X 2 = (1) 5 8 + (4) 3 8 = ( - 1) 2 1 4 + (1) 2 3 8 + ( - 2) 2 1 8 + (2) 2 1 4 . So even in this case E X 2 = x x 2 P ( X = x ). Theorem 4.2. E g ( X ) = g ( x ) p ( x ) . Proof. Let Y = g ( X ). Then E Y = X y y P ( Y = y ) = X y y X { x : g ( x )= y } P ( X = x ) = X x g ( x ) P ( X = x ) . Example. E X 2 = x 2 p ( x ). E X n is called the n th moment of X . If M = E X , then Var ( X ) = E ( X - M ) 2 is called the variance of X . The square root of Var (
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Unformatted text preview: X ) is the standard deviation of X . The variance measures how much spread there is about the expected value. Example. We toss a fair coin and let X = 1 if we get heads, X =-1 if we get tails. Then E X = 0, so X-E X = X , and then Var X = E X 2 = (1) 2 1 2 + (-1) 2 1 2 = 1. Example. We roll a die and let X be the value that shows. We have previously calculated E X = 7 2 . So X-E X equals-5 2 ,-3 2 ,-1 2 , 1 2 , 3 2 , 5 2 , each with probability 1 6 . So Var X = (-5 2 ) 2 1 6 + (-3 2 ) 2 1 6 + (-1 2 ) 2 1 6 + ( 1 2 ) 2 1 6 + ( 3 2 ) 2 1 6 + ( 5 2 ) 2 1 6 = 35 12 . Note that the expectation of a constant is just the constant. An alternate expression for the variance is Var X = E X 2-2 E ( XM ) + E ( M 2 ) = E X 2-2 M 2 + M 2 = E X 2-( E X ) 2 . 11...
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