Unformatted text preview: (−c). Thus,
FY (c) = FX (c) − FX (−c) c ≥ 0
c ≤ 0; Diﬀerentiating to get the pdf yields:
fY (c) = fX (c) + fX (−c) c ≥ 0
c < 0; Basically, for each c > 0, there are two terms in the expression for fY (c) because there are two ways
for Y to be c–either X = c or X = −c. A geometric interpretation is given in Figure 3.19. Figure
3.19(a) pictures a possible pdf for X. Figure 3.19(b) shows a decomposition of the probability mass
into a part on the negative line and part on the positive line. Figure 3.19(c) shows the result of
reﬂecting the probability mass on the negative line to the positive line. Figure 3.19(d) shows the
pdf of |Y |, obtained by adding the two functions in Figure 3.19(c).
Another way to think of this geometrically would be to fold a picture of the pdf of X in half
along the vertical axis, so that fX (c) and fX (−c) are lined up for each c, and then add these
together to get fY (c) for c ≥ 0.. Example 3.8.9 Let X be an exponentially distributed random variable with parameter λ. Let...
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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
- The Land