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Unformatted text preview: is 1/(2a(1 − a)). So
the mean number of coin ﬂips required to produce one Bernoulli random variable with parameter
0.5 using the biased coin by this method is 1/(a(1 − a)). 3.9. FAILURE RATE FUNCTIONS 111
u
u=1
!1
F (u)
X F (c)
X +
F (c)
X ! c
0 Figure 3.23: E [X ] is the area of the + region minus the area of the − region. 3.8.3 The area rule for expectation based on the CDF There is a simple rule for determining the expectation, E [X ], of a random variable X directly from
its CDF, called the area rule for expectation. See Figure 3.23, which shows an example of a CDF
FX plotted as a function of c in the c − u plane. Consider the inﬁnite strip bounded by the c axis
and the horizontal line given by u = 1. Then E [X ] is the area of the region in the strip to the right
of the u axis above the CDF, minus the area of the region in the strip to the left of the u axis
below the CDF, as long as at least one of the two regions has ﬁnite area. The area rule can also be
written as an...
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 Spring '08
 Zahrn
 The Land

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