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Unformatted text preview: a) Express fY in terms of fX .
(b) Sketch fY in the case X is uniformly distributed over the interval [15, 20].
Solution: (a) By the scaling formula with a = 1.8 and b = 32, fY (c) = fX ( c−.32 )/1.8.
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(b) The case when X is uniformly distributed over [15, 20] leads to Y uniformly distributed over
[59, 68]. This is illustrated in Figure 3.9. The pdf of X is shown at the top of the ﬁgure. The pdf
of (1.8)X is shown in the middle of the ﬁgure. It is obtained by stretching fX out from the origin
by a factor 1.8, and at the same time reducing the height by the same factor, so the area under the
curve is still one. Finally, fY is obtained by sliding the pdf of (1.8)X to the right by 32. 88 CHAPTER 3. CONTINUOUSTYPE RANDOM VARIABLES
fx
fy
0 10 20 30 40 50 60 70 Figure 3.9: Rescaling from degrees C to degrees F. Example 3.6.2 Let T denote the duration of a waiting time in a service system, measured in
seconds. Suppose T is exponentially distributed with parameter λ = 0.01. Let S denote the same
waiting time, but measured in minutes. Find the mean and the pdf of S.
Solution: First we identify the pdf...
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 Spring '08
 Zahrn
 The Land

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