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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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 Fall '09

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