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P {− ln(1 − U ) ≤ c} = P {ln(1 − U ) ≥ −c} = P {1 − U ≥ e−c } = P {U ≤ 1 − e−c } = F (c).
The choice of g is not unique in general. For example, 1 − U has the same distribution as U , so the
CDF of − ln(U ) is also F . Example 3.8.13 Find a function g so that, if U is uniformly distributed over the interval [0, 1],
then g (U ) has the distribution of the number showing for the experiment of rolling a fair die.
Solution: The desired CDF of g (U ) is shown in Figure 3.1. Using g = F −1 and using (3.7) or
the graphical method illustrated in Figure 3.22 to ﬁnd F −1 , we get that for 0 < u < 1, g (u) = i for
i−1
i
6 < u ≤ 6 for 1 ≤ i ≤ 6. To double check the answer, note that if 1 ≤ i ≤ 6, then
P {g (U ) = i } = P i
i−1
<U ≤
6
6 1
=,
6 so g (U ) has the correct pmf, and hence the correct CDF. Example 3.8.14 (This example is a puzzle that doesn’t use the theory of this section.) We’ve
discussed how to generate a random variable with a speciﬁed distribution starting with an uniformly
distributed random variable. Suppose instea...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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