Pfalse alarm p x h0 m0 x m0 p h0 m0 q pmiss

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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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