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Unformatted text preview: ift the graph horizontally by b (to the right if b > 0 or to the left if b < 0.)
Here is a derivation of the scaling rule (3.3). Express the CDF of Y as:
FY (v ) = P {aX + b ≤ v } = P X≤ v−b
a = FX v−b
a , and then diﬀerentiate FY (v ) with respect to v , using the chain rule of calculus and the fact FX = fX ,
to obtain:
v−b 1
fY (v ) = FY (v ) = fX
.
a
a
Section 3.2 recounts how the mean, variance, and standard deviation of Y are related to the
mean, variance, and standard deviation of X, in case Y = aX + b. These relations are the same
ones discussed in Section 2.2 for discretetype random variables, namely:
E [Y ] = aE [X ] + b Var(Y) = a2 Var(X ) σY = aσX . Example 3.6.1 Let X denote the pdf of the high temperature, in degrees C (Celsius), for a
certain day of the year in some city. Let Y denote the pdf of the same temperature, but in degrees
F (Fahrenheit). The conversion formula is Y = (1.8)X + 32. This is the linear transformation that
maps zero degrees C to 32 degrees F and 100 degrees C to 212 degrees F.
(...
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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.
 Spring '08
 Zahrn
 The Land

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