Equivalently ln1 f h or ln1 f ln1 f 0 0

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Unformatted text preview: vertical interval is equal to the probability that X falls into the horizontal interval on the u axis. The approximate ratio of the length of the vertical interval to the length of the horizontal interval is g (g −1 (c)). The density fX (g −1 (c)) is divided by this ratio in (3.6). Assuming Figure 3.21 is drawn to scale, the derivative of g at g −1 (c) is about 1/4. Therefore the density of Y at c is about four times larger than the density of X at g −1 (c). Example 3.8.11 Suppose X is a continuous-type random variable with CDF FX . Let Y be the result of applying FX to X, that is, Y = FX (X ). Find the distribution of Y. Solution: Since Y takes values in the interval [0, 1], let 0 < v < 1. Since F increases continuously from zero to one, there is a value cv such that FX (cv ) = v. Then P {FX (X ) ≤ v } = P {X ≤ cv } = F (cv ) = v. That is, FX (X ) is uniformly distributed over the interval [0, 1]. This result may seem surprising at first, but it is natural if it is thought...
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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.

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