elemprob-fall2010-page28

elemprob-fall2010-page28 - Var X = E X 2-( E X ) 2 = ( b-a...

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Proposition 11.1 E g ( X ) = R g ( x ) f ( x ) dx. As in the discrete case, Var X = E [ X - E X ] 2 . As an example of these calculations, let us look at the uniform distribution. We say that a random variable X has a uniform distribution on [ a,b ] if f X ( x ) = 1 b - a if a x b and 0 otherwise. To calculate the expectation of X , E X = Z -∞ xf X ( x ) dx = Z b a x 1 b - a dx = 1 b - a Z b a xdx = 1 b - a ± b 2 2 - a 2 2 ² = a + b 2 . This is what one would expect. To calculate the variance, we first calculate E X 2 = Z -∞ x 2 f X ( x ) dx = Z b a x 2 1 b - a dx = a 2 + ab + b 2 3 . We then do some algebra to obtain
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Unformatted text preview: Var X = E X 2-( E X ) 2 = ( b-a ) 2 12 . 12 Normal distribution A random variable is a standard normal (written N (0 , 1)) if it has density 1 2 e-x 2 / 2 . A synonym for normal is Gaussian. The rst thing to do is show that this is a density. Let I = R e-x 2 / 2 dx. Then I 2 = Z Z - e-x 2 / 2 e-y 2 / 2 dxdy. 28...
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