Suppose f1 and f0 are two known pdfs and suppose that

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Unformatted text preview: y real co and uo with 0 < uo < 1, !1 F (u) F(c) 1 u c 1 Figure 3.22: A CDF and its inverse. F −1 (uo ) ≤ co if and only if uo ≤ F (co ). Thus, if X = F −1 (U ) then FX (c) = P {F −1 (U ) ≤ c} = P {U ≤ F (c)} = F (c), so indeed F is the CDF of X. Example 3.8.12 Find a function g so that, if U is uniformly distributed over the interval [0, 1], g (U ) is exponentially distributed with parameter λ = 1. Solution: The desired exponential distribution has support R+ and CDF F given by: F (c) = 1 − e−c for c ≥ 0 and F (c) = 0 for c < 0. We’ll let g (u) = F −1 (u). Since F is strictly and continuously increasing over the support, if 0 < u < 1 then the value c of F −1 (u) is such that 110 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES F (c) = u. That is, we would like 1 − e−c = u which is equivalent to e−c = 1 − u, or c = − ln(1 − u). Thus, F −1 (u) = − ln(1 − u). So we can take g (u) = − ln(1 − u) for 0 < u < 1. To double check the answer, note that if c ≥ 0, th...
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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 Tech.

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