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This also makes sense with the denitio probability as

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Unformatted text preview: ty as area; there is no area between 180 cm to 180 cm. 21 ORF 245 Prof. Rigollet Fall 2012 Expected Value The expected value of a continuous random ¯ variable is also the limit of x as the number of observations goes to infinity In this case, we have µ = E (X ) = ⇥ xf (x)dx ⇥ For a general function h(X ) we have E [h(X )] = ⇥ ⇥ h(x)f (x)dx ORF 245 Prof. Rigollet Fall 2012 Variance The variance of a continuous random variable is obtained by taking h(X ) = (X µ)2 var(X ) = ⇥ (x µ)2 f (x)dx ⇥ We still have the shortcut formula var(X ) = E (X 2 ) µ2 = ⇤ x2 f (x)dx ⇤ ⇥2 xf (x)dx Moreover, the variance is the limit of s2 as the number of observations goes to infinity. ORF 245 Prof. Rigollet Fall 2012 Rules If the function h(X) is of the form h(X)=aX+b for some numbers a and b, we have other shortcuts: E (aX + b) = aE (X ) + b var(aX + b) = a2 var(X ) These rules apply whether the random variable is discrete or continuous. Summary ORF 245 Prof. Rigollet Fall 2012 Probability: ● Random experiment => Random variable X. ● Possible values for X = elementary outcomes. ● Event = collection of elementary outcomes. ● P(X=x) is the probability that the realization of the random variable X is the elementary outcome x. ● Rules to compute P(event). ● µ = E (X ) is the expected value of X (we also defined the variance). X Statistics: x · (number of xi equal to x) ● data x1 , x2 , . . . , xn = realization of n independent random possible X experiments (with corresponding random variable X). x · (proportion equal to tends to P(X=x). n ● LLN: proportion of xi equal to x x) 1 possibleSample mean (see also the sample variance): x = ¯ xi ● n i=1...
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