To show that the normal density integrates to one it

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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. 18 (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. CONTINUOUS-TYPE 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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