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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 continuoustype 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.
 Spring '08
 Zahrn
 The Land

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