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function mapping the interval (a, b) onto the interval (A, B ). The situation is illustrated in Figure
3.21. The support of Y is the interval [A, B ]. So let A < c < B. There is a value g −1 (c) on the u
axis such that g (g −1 (c)) = c, and:
FY (c) = P {Y ≤ c} = P {X ≤ g −1 (c)} = FX (g −1 (c)).
The derivative of the inverse of a function is one over the derivative of the function itself4 , to yield
1
g −1 (c) = g (g−1 (c)) . Thus, diﬀerentiating FY yields:
fY (c) =
4 1
fX (g −1 (c)) g (g−1 (c))
0 A<c<B
else. Prove this by diﬀerentiating both sides of the identity g (g −1 (c)) = c (3.6) 108 CHAPTER 3. CONTINUOUSTYPE RANDOM VARIABLES B
c
g(u)
A
f (u)
X
u
a !1 g (c) b Figure 3.21: Monotone function of a continuoustype random variable.
The expression for fY in (3.6) has an appealing form. Figure 3.21 shows a small interval with one
endpoint c on the vertical axis, and the inverse image of the interval on the u axis, which is a
small interval with one endpoint g −1 (c). The probability Y falls into the...
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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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